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<h1>Source code for networkx.algorithms.approximation.traveling_salesman</h1><div class="highlight"><pre>
<span></span><span class="sd">"""</span>
<span class="sd">=================================</span>
<span class="sd">Travelling Salesman Problem (TSP)</span>
<span class="sd">=================================</span>
<span class="sd">Implementation of approximate algorithms</span>
<span class="sd">for solving and approximating the TSP problem.</span>
<span class="sd">Categories of algorithms which are implemented:</span>
<span class="sd">- Christofides (provides a 3/2-approximation of TSP)</span>
<span class="sd">- Greedy</span>
<span class="sd">- Simulated Annealing (SA)</span>
<span class="sd">- Threshold Accepting (TA)</span>
<span class="sd">- Asadpour Asymmetric Traveling Salesman Algorithm</span>
<span class="sd">The Travelling Salesman Problem tries to find, given the weight</span>
<span class="sd">(distance) between all points where a salesman has to visit, the</span>
<span class="sd">route so that:</span>
<span class="sd">- The total distance (cost) which the salesman travels is minimized.</span>
<span class="sd">- The salesman returns to the starting point.</span>
<span class="sd">- Note that for a complete graph, the salesman visits each point once.</span>
<span class="sd">The function `travelling_salesman_problem` allows for incomplete</span>
<span class="sd">graphs by finding all-pairs shortest paths, effectively converting</span>
<span class="sd">the problem to a complete graph problem. It calls one of the</span>
<span class="sd">approximate methods on that problem and then converts the result</span>
<span class="sd">back to the original graph using the previously found shortest paths.</span>
<span class="sd">TSP is an NP-hard problem in combinatorial optimization,</span>
<span class="sd">important in operations research and theoretical computer science.</span>
<span class="sd">http://en.wikipedia.org/wiki/Travelling_salesman_problem</span>
<span class="sd">"""</span>
<span class="kn">import</span> <span class="nn">math</span>
<span class="kn">import</span> <span class="nn">networkx</span> <span class="k">as</span> <span class="nn">nx</span>
<span class="kn">from</span> <span class="nn">networkx.algorithms.tree.mst</span> <span class="kn">import</span> <span class="n">random_spanning_tree</span>
<span class="kn">from</span> <span class="nn">networkx.utils</span> <span class="kn">import</span> <span class="n">not_implemented_for</span><span class="p">,</span> <span class="n">pairwise</span><span class="p">,</span> <span class="n">py_random_state</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
<span class="s2">"traveling_salesman_problem"</span><span class="p">,</span>
<span class="s2">"christofides"</span><span class="p">,</span>
<span class="s2">"asadpour_atsp"</span><span class="p">,</span>
<span class="s2">"greedy_tsp"</span><span class="p">,</span>
<span class="s2">"simulated_annealing_tsp"</span><span class="p">,</span>
<span class="s2">"threshold_accepting_tsp"</span><span class="p">,</span>
<span class="p">]</span>
<span class="k">def</span> <span class="nf">swap_two_nodes</span><span class="p">(</span><span class="n">soln</span><span class="p">,</span> <span class="n">seed</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Swap two nodes in `soln` to give a neighbor solution.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> soln : list of nodes</span>
<span class="sd"> Current cycle of nodes</span>
<span class="sd"> seed : integer, random_state, or None (default)</span>
<span class="sd"> Indicator of random number generation state.</span>
<span class="sd"> See :ref:`Randomness<randomness>`.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> list</span>
<span class="sd"> The solution after move is applied. (A neighbor solution.)</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This function assumes that the incoming list `soln` is a cycle</span>
<span class="sd"> (that the first and last element are the same) and also that</span>
<span class="sd"> we don't want any move to change the first node in the list</span>
<span class="sd"> (and thus not the last node either).</span>
<span class="sd"> The input list is changed as well as returned. Make a copy if needed.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> move_one_node</span>
<span class="sd"> """</span>
<span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">seed</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">soln</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">k</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">soln</span><span class="p">[</span><span class="n">a</span><span class="p">],</span> <span class="n">soln</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">=</span> <span class="n">soln</span><span class="p">[</span><span class="n">b</span><span class="p">],</span> <span class="n">soln</span><span class="p">[</span><span class="n">a</span><span class="p">]</span>
<span class="k">return</span> <span class="n">soln</span>
<span class="k">def</span> <span class="nf">move_one_node</span><span class="p">(</span><span class="n">soln</span><span class="p">,</span> <span class="n">seed</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Move one node to another position to give a neighbor solution.</span>
<span class="sd"> The node to move and the position to move to are chosen randomly.</span>
<span class="sd"> The first and last nodes are left untouched as soln must be a cycle</span>
<span class="sd"> starting at that node.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> soln : list of nodes</span>
<span class="sd"> Current cycle of nodes</span>
<span class="sd"> seed : integer, random_state, or None (default)</span>
<span class="sd"> Indicator of random number generation state.</span>
<span class="sd"> See :ref:`Randomness<randomness>`.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> list</span>
<span class="sd"> The solution after move is applied. (A neighbor solution.)</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This function assumes that the incoming list `soln` is a cycle</span>
<span class="sd"> (that the first and last element are the same) and also that</span>
<span class="sd"> we don't want any move to change the first node in the list</span>
<span class="sd"> (and thus not the last node either).</span>
<span class="sd"> The input list is changed as well as returned. Make a copy if needed.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> swap_two_nodes</span>
<span class="sd"> """</span>
<span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">seed</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">soln</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">k</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">soln</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">soln</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
<span class="k">return</span> <span class="n">soln</span>
<div class="viewcode-block" id="christofides"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.approximation.traveling_salesman.christofides.html#networkx.algorithms.approximation.traveling_salesman.christofides">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"directed"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">christofides</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">,</span> <span class="n">tree</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Approximate a solution of the traveling salesman problem</span>
<span class="sd"> Compute a 3/2-approximation of the traveling salesman problem</span>
<span class="sd"> in a complete undirected graph using Christofides [1]_ algorithm.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : Graph</span>
<span class="sd"> `G` should be a complete weighted undirected graph.</span>
<span class="sd"> The distance between all pairs of nodes should be included.</span>
<span class="sd"> weight : string, optional (default="weight")</span>
<span class="sd"> Edge data key corresponding to the edge weight.</span>
<span class="sd"> If any edge does not have this attribute the weight is set to 1.</span>
<span class="sd"> tree : NetworkX graph or None (default: None)</span>
<span class="sd"> A minimum spanning tree of G. Or, if None, the minimum spanning</span>
<span class="sd"> tree is computed using :func:`networkx.minimum_spanning_tree`</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> list</span>
<span class="sd"> List of nodes in `G` along a cycle with a 3/2-approximation of</span>
<span class="sd"> the minimal Hamiltonian cycle.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Christofides, Nicos. "Worst-case analysis of a new heuristic for</span>
<span class="sd"> the travelling salesman problem." No. RR-388. Carnegie-Mellon Univ</span>
<span class="sd"> Pittsburgh Pa Management Sciences Research Group, 1976.</span>
<span class="sd"> """</span>
<span class="c1"># Remove selfloops if necessary</span>
<span class="n">loop_nodes</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">nodes_with_selfloops</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">node</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">loop_nodes</span><span class="p">)</span>
<span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
<span class="k">pass</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">G</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">G</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">node</span><span class="p">,</span> <span class="n">node</span><span class="p">)</span>
<span class="n">G</span><span class="o">.</span><span class="n">remove_edges_from</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">loop_nodes</span><span class="p">)</span>
<span class="c1"># Check that G is a complete graph</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
<span class="c1"># This check ignores selfloops which is what we want here.</span>
<span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nbrdict</span><span class="p">)</span> <span class="o">!=</span> <span class="n">N</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">nbrdict</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">adj</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"G must be a complete graph."</span><span class="p">)</span>
<span class="k">if</span> <span class="n">tree</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">tree</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">minimum_spanning_tree</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span>
<span class="n">L</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">L</span><span class="o">.</span><span class="n">remove_nodes_from</span><span class="p">([</span><span class="n">v</span> <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">degree</span> <span class="ow">in</span> <span class="n">tree</span><span class="o">.</span><span class="n">degree</span> <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">degree</span> <span class="o">%</span> <span class="mi">2</span><span class="p">)])</span>
<span class="n">MG</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">MultiGraph</span><span class="p">()</span>
<span class="n">MG</span><span class="o">.</span><span class="n">add_edges_from</span><span class="p">(</span><span class="n">tree</span><span class="o">.</span><span class="n">edges</span><span class="p">)</span>
<span class="n">edges</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">min_weight_matching</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span>
<span class="n">MG</span><span class="o">.</span><span class="n">add_edges_from</span><span class="p">(</span><span class="n">edges</span><span class="p">)</span>
<span class="k">return</span> <span class="n">_shortcutting</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">eulerian_circuit</span><span class="p">(</span><span class="n">MG</span><span class="p">))</span></div>
<span class="k">def</span> <span class="nf">_shortcutting</span><span class="p">(</span><span class="n">circuit</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Remove duplicate nodes in the path"""</span>
<span class="n">nodes</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">circuit</span><span class="p">:</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">:</span>
<span class="k">continue</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nodes</span><span class="p">:</span>
<span class="n">nodes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
<span class="n">nodes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="n">nodes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">nodes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="k">return</span> <span class="n">nodes</span>
<div class="viewcode-block" id="traveling_salesman_problem"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.approximation.traveling_salesman.traveling_salesman_problem.html#networkx.algorithms.approximation.traveling_salesman.traveling_salesman_problem">[docs]</a><span class="k">def</span> <span class="nf">traveling_salesman_problem</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">,</span> <span class="n">nodes</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">cycle</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Find the shortest path in `G` connecting specified nodes</span>
<span class="sd"> This function allows approximate solution to the traveling salesman</span>
<span class="sd"> problem on networks that are not complete graphs and/or where the</span>
<span class="sd"> salesman does not need to visit all nodes.</span>
<span class="sd"> This function proceeds in two steps. First, it creates a complete</span>
<span class="sd"> graph using the all-pairs shortest_paths between nodes in `nodes`.</span>
<span class="sd"> Edge weights in the new graph are the lengths of the paths</span>
<span class="sd"> between each pair of nodes in the original graph.</span>
<span class="sd"> Second, an algorithm (default: `christofides` for undirected and</span>
<span class="sd"> `asadpour_atsp` for directed) is used to approximate the minimal Hamiltonian</span>
<span class="sd"> cycle on this new graph. The available algorithms are:</span>
<span class="sd"> - christofides</span>
<span class="sd"> - greedy_tsp</span>
<span class="sd"> - simulated_annealing_tsp</span>
<span class="sd"> - threshold_accepting_tsp</span>
<span class="sd"> - asadpour_atsp</span>
<span class="sd"> Once the Hamiltonian Cycle is found, this function post-processes to</span>
<span class="sd"> accommodate the structure of the original graph. If `cycle` is ``False``,</span>
<span class="sd"> the biggest weight edge is removed to make a Hamiltonian path.</span>
<span class="sd"> Then each edge on the new complete graph used for that analysis is</span>
<span class="sd"> replaced by the shortest_path between those nodes on the original graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> A possibly weighted graph</span>
<span class="sd"> nodes : collection of nodes (default=G.nodes)</span>
<span class="sd"> collection (list, set, etc.) of nodes to visit</span>
<span class="sd"> weight : string, optional (default="weight")</span>
<span class="sd"> Edge data key corresponding to the edge weight.</span>
<span class="sd"> If any edge does not have this attribute the weight is set to 1.</span>
<span class="sd"> cycle : bool (default: True)</span>
<span class="sd"> Indicates whether a cycle should be returned, or a path.</span>
<span class="sd"> Note: the cycle is the approximate minimal cycle.</span>
<span class="sd"> The path simply removes the biggest edge in that cycle.</span>
<span class="sd"> method : function (default: None)</span>
<span class="sd"> A function that returns a cycle on all nodes and approximates</span>
<span class="sd"> the solution to the traveling salesman problem on a complete</span>
<span class="sd"> graph. The returned cycle is then used to find a corresponding</span>
<span class="sd"> solution on `G`. `method` should be callable; take inputs</span>
<span class="sd"> `G`, and `weight`; and return a list of nodes along the cycle.</span>
<span class="sd"> Provided options include :func:`christofides`, :func:`greedy_tsp`,</span>
<span class="sd"> :func:`simulated_annealing_tsp` and :func:`threshold_accepting_tsp`.</span>
<span class="sd"> If `method is None`: use :func:`christofides` for undirected `G` and</span>
<span class="sd"> :func:`threshold_accepting_tsp` for directed `G`.</span>
<span class="sd"> To specify parameters for these provided functions, construct lambda</span>
<span class="sd"> functions that state the specific value. `method` must have 2 inputs.</span>
<span class="sd"> (See examples).</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> list</span>
<span class="sd"> List of nodes in `G` along a path with an approximation of the minimal</span>
<span class="sd"> path through `nodes`.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If `G` is a directed graph it has to be strongly connected or the</span>
<span class="sd"> complete version cannot be generated.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> tsp = nx.approximation.traveling_salesman_problem</span>
<span class="sd"> >>> G = nx.cycle_graph(9)</span>
<span class="sd"> >>> G[4][5]["weight"] = 5 # all other weights are 1</span>
<span class="sd"> >>> tsp(G, nodes=[3, 6])</span>
<span class="sd"> [3, 2, 1, 0, 8, 7, 6, 7, 8, 0, 1, 2, 3]</span>
<span class="sd"> >>> path = tsp(G, cycle=False)</span>
<span class="sd"> >>> path in ([4, 3, 2, 1, 0, 8, 7, 6, 5], [5, 6, 7, 8, 0, 1, 2, 3, 4])</span>
<span class="sd"> True</span>
<span class="sd"> Build (curry) your own function to provide parameter values to the methods.</span>
<span class="sd"> >>> SA_tsp = nx.approximation.simulated_annealing_tsp</span>
<span class="sd"> >>> method = lambda G, wt: SA_tsp(G, "greedy", weight=wt, temp=500)</span>
<span class="sd"> >>> path = tsp(G, cycle=False, method=method)</span>
<span class="sd"> >>> path in ([4, 3, 2, 1, 0, 8, 7, 6, 5], [5, 6, 7, 8, 0, 1, 2, 3, 4])</span>
<span class="sd"> True</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">method</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="n">method</span> <span class="o">=</span> <span class="n">asadpour_atsp</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">method</span> <span class="o">=</span> <span class="n">christofides</span>
<span class="k">if</span> <span class="n">nodes</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">nodes</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">nodes</span><span class="p">)</span>
<span class="n">dist</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">path</span> <span class="o">=</span> <span class="p">{}</span>
<span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">all_pairs_dijkstra</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">):</span>
<span class="n">dist</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">d</span>
<span class="n">path</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">p</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="c1"># If the graph is not strongly connected, raise an exception</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_strongly_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"G is not strongly connected"</span><span class="p">)</span>
<span class="n">GG</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">DiGraph</span><span class="p">()</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">GG</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">Graph</span><span class="p">()</span>
<span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">:</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">:</span>
<span class="k">if</span> <span class="n">u</span> <span class="o">==</span> <span class="n">v</span><span class="p">:</span>
<span class="k">continue</span>
<span class="n">GG</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">dist</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">])</span>
<span class="n">best_GG</span> <span class="o">=</span> <span class="n">method</span><span class="p">(</span><span class="n">GG</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">cycle</span><span class="p">:</span>
<span class="c1"># find and remove the biggest edge</span>
<span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">pairwise</span><span class="p">(</span><span class="n">best_GG</span><span class="p">),</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">dist</span><span class="p">[</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]][</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]])</span>
<span class="n">pos</span> <span class="o">=</span> <span class="n">best_GG</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
<span class="k">while</span> <span class="n">best_GG</span><span class="p">[</span><span class="n">pos</span><span class="p">]</span> <span class="o">!=</span> <span class="n">v</span><span class="p">:</span>
<span class="n">pos</span> <span class="o">=</span> <span class="n">best_GG</span><span class="p">[</span><span class="n">pos</span><span class="p">:]</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
<span class="n">best_GG</span> <span class="o">=</span> <span class="n">best_GG</span><span class="p">[</span><span class="n">pos</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">best_GG</span><span class="p">[:</span><span class="n">pos</span><span class="p">]</span>
<span class="n">best_path</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">pairwise</span><span class="p">(</span><span class="n">best_GG</span><span class="p">):</span>
<span class="n">best_path</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">path</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">][:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
<span class="n">best_path</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="k">return</span> <span class="n">best_path</span></div>
<div class="viewcode-block" id="asadpour_atsp"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.approximation.traveling_salesman.asadpour_atsp.html#networkx.algorithms.approximation.traveling_salesman.asadpour_atsp">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="nd">@py_random_state</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">asadpour_atsp</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">source</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Returns an approximate solution to the traveling salesman problem.</span>
<span class="sd"> This approximate solution is one of the best known approximations for the</span>
<span class="sd"> asymmetric traveling salesman problem developed by Asadpour et al,</span>
<span class="sd"> [1]_. The algorithm first solves the Held-Karp relaxation to find a lower</span>
<span class="sd"> bound for the weight of the cycle. Next, it constructs an exponential</span>
<span class="sd"> distribution of undirected spanning trees where the probability of an</span>
<span class="sd"> edge being in the tree corresponds to the weight of that edge using a</span>
<span class="sd"> maximum entropy rounding scheme. Next we sample that distribution</span>
<span class="sd"> $2 \\lceil \\ln n \\rceil$ times and save the minimum sampled tree once the</span>
<span class="sd"> direction of the arcs is added back to the edges. Finally, we augment</span>
<span class="sd"> then short circuit that graph to find the approximate tour for the</span>
<span class="sd"> salesman.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : nx.DiGraph</span>
<span class="sd"> The graph should be a complete weighted directed graph. The</span>
<span class="sd"> distance between all paris of nodes should be included and the triangle</span>
<span class="sd"> inequality should hold. That is, the direct edge between any two nodes</span>
<span class="sd"> should be the path of least cost.</span>
<span class="sd"> weight : string, optional (default="weight")</span>
<span class="sd"> Edge data key corresponding to the edge weight.</span>
<span class="sd"> If any edge does not have this attribute the weight is set to 1.</span>
<span class="sd"> seed : integer, random_state, or None (default)</span>
<span class="sd"> Indicator of random number generation state.</span>
<span class="sd"> See :ref:`Randomness<randomness>`.</span>
<span class="sd"> source : node label (default=`None`)</span>
<span class="sd"> If given, return the cycle starting and ending at the given node.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> cycle : list of nodes</span>
<span class="sd"> Returns the cycle (list of nodes) that a salesman can follow to minimize</span>
<span class="sd"> the total weight of the trip.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If `G` is not complete or has less than two nodes, the algorithm raises</span>
<span class="sd"> an exception.</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If `source` is not `None` and is not a node in `G`, the algorithm raises</span>
<span class="sd"> an exception.</span>
<span class="sd"> NetworkXNotImplemented</span>
<span class="sd"> If `G` is an undirected graph.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] A. Asadpour, M. X. Goemans, A. Madry, S. O. Gharan, and A. Saberi,</span>
<span class="sd"> An o(log n/log log n)-approximation algorithm for the asymmetric</span>
<span class="sd"> traveling salesman problem, Operations research, 65 (2017),</span>
<span class="sd"> pp. 1043–1061</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> import networkx as nx</span>
<span class="sd"> >>> import networkx.algorithms.approximation as approx</span>
<span class="sd"> >>> G = nx.complete_graph(3, create_using=nx.DiGraph)</span>
<span class="sd"> >>> nx.set_edge_attributes(G, {(0, 1): 2, (1, 2): 2, (2, 0): 2, (0, 2): 1, (2, 1): 1, (1, 0): 1}, "weight")</span>
<span class="sd"> >>> tour = approx.asadpour_atsp(G,source=0)</span>
<span class="sd"> >>> tour</span>
<span class="sd"> [0, 2, 1, 0]</span>
<span class="sd"> """</span>
<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">ceil</span><span class="p">,</span> <span class="n">exp</span>
<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">log</span> <span class="k">as</span> <span class="n">ln</span>
<span class="c1"># Check that G is a complete graph</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
<span class="k">if</span> <span class="n">N</span> <span class="o"><</span> <span class="mi">2</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"G must have at least two nodes"</span><span class="p">)</span>
<span class="c1"># This check ignores selfloops which is what we want here.</span>
<span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nbrdict</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">n</span> <span class="ow">in</span> <span class="n">nbrdict</span><span class="p">)</span> <span class="o">!=</span> <span class="n">N</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">nbrdict</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">adj</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"G is not a complete DiGraph"</span><span class="p">)</span>
<span class="c1"># Check that the source vertex, if given, is in the graph</span>
<span class="k">if</span> <span class="n">source</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="ow">and</span> <span class="n">source</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">nodes</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"Given source node not in G."</span><span class="p">)</span>
<span class="n">opt_hk</span><span class="p">,</span> <span class="n">z_star</span> <span class="o">=</span> <span class="n">held_karp_ascent</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="c1"># Test to see if the ascent method found an integer solution or a fractional</span>
<span class="c1"># solution. If it is integral then z_star is a nx.Graph, otherwise it is</span>
<span class="c1"># a dict</span>
<span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">z_star</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
<span class="c1"># Here we are using the shortcutting method to go from the list of edges</span>
<span class="c1"># returned from eularian_circuit to a list of nodes</span>
<span class="k">return</span> <span class="n">_shortcutting</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">eulerian_circuit</span><span class="p">(</span><span class="n">z_star</span><span class="p">,</span> <span class="n">source</span><span class="o">=</span><span class="n">source</span><span class="p">))</span>
<span class="c1"># Create the undirected support of z_star</span>
<span class="n">z_support</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">MultiGraph</span><span class="p">()</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">z_star</span><span class="p">:</span>
<span class="k">if</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">z_support</span><span class="o">.</span><span class="n">edges</span><span class="p">:</span>
<span class="n">edge_weight</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">][</span><span class="n">weight</span><span class="p">],</span> <span class="n">G</span><span class="p">[</span><span class="n">v</span><span class="p">][</span><span class="n">u</span><span class="p">][</span><span class="n">weight</span><span class="p">])</span>
<span class="n">z_support</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="n">weight</span><span class="p">:</span> <span class="n">edge_weight</span><span class="p">})</span>
<span class="c1"># Create the exponential distribution of spanning trees</span>
<span class="n">gamma</span> <span class="o">=</span> <span class="n">spanning_tree_distribution</span><span class="p">(</span><span class="n">z_support</span><span class="p">,</span> <span class="n">z_star</span><span class="p">)</span>
<span class="c1"># Write the lambda values to the edges of z_support</span>
<span class="n">z_support</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">Graph</span><span class="p">(</span><span class="n">z_support</span><span class="p">)</span>
<span class="n">lambda_dict</span> <span class="o">=</span> <span class="p">{(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span> <span class="n">exp</span><span class="p">(</span><span class="n">gamma</span><span class="p">[(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)])</span> <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">z_support</span><span class="o">.</span><span class="n">edges</span><span class="p">()}</span>
<span class="n">nx</span><span class="o">.</span><span class="n">set_edge_attributes</span><span class="p">(</span><span class="n">z_support</span><span class="p">,</span> <span class="n">lambda_dict</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">)</span>
<span class="k">del</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">lambda_dict</span>
<span class="c1"># Sample 2 * ceil( ln(n) ) spanning trees and record the minimum one</span>
<span class="n">minimum_sampled_tree</span> <span class="o">=</span> <span class="kc">None</span>
<span class="n">minimum_sampled_tree_weight</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">inf</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">ceil</span><span class="p">(</span><span class="n">ln</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">number_of_nodes</span><span class="p">()))):</span>
<span class="n">sampled_tree</span> <span class="o">=</span> <span class="n">random_spanning_tree</span><span class="p">(</span><span class="n">z_support</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="n">seed</span><span class="p">)</span>
<span class="n">sampled_tree_weight</span> <span class="o">=</span> <span class="n">sampled_tree</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">weight</span><span class="p">)</span>
<span class="k">if</span> <span class="n">sampled_tree_weight</span> <span class="o"><</span> <span class="n">minimum_sampled_tree_weight</span><span class="p">:</span>
<span class="n">minimum_sampled_tree</span> <span class="o">=</span> <span class="n">sampled_tree</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">minimum_sampled_tree_weight</span> <span class="o">=</span> <span class="n">sampled_tree_weight</span>
<span class="c1"># Orient the edges in that tree to keep the cost of the tree the same.</span>
<span class="n">t_star</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">MultiDiGraph</span><span class="p">()</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">minimum_sampled_tree</span><span class="o">.</span><span class="n">edges</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">weight</span><span class="p">):</span>
<span class="k">if</span> <span class="n">d</span> <span class="o">==</span> <span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">][</span><span class="n">weight</span><span class="p">]:</span>
<span class="n">t_star</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="n">weight</span><span class="p">:</span> <span class="n">d</span><span class="p">})</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">t_star</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="n">weight</span><span class="p">:</span> <span class="n">d</span><span class="p">})</span>
<span class="c1"># Find the node demands needed to neutralize the flow of t_star in G</span>
<span class="n">node_demands</span> <span class="o">=</span> <span class="p">{</span><span class="n">n</span><span class="p">:</span> <span class="n">t_star</span><span class="o">.</span><span class="n">out_degree</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">-</span> <span class="n">t_star</span><span class="o">.</span><span class="n">in_degree</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">t_star</span><span class="p">}</span>
<span class="n">nx</span><span class="o">.</span><span class="n">set_node_attributes</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">node_demands</span><span class="p">,</span> <span class="s2">"demand"</span><span class="p">)</span>
<span class="c1"># Find the min_cost_flow</span>
<span class="n">flow_dict</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">min_cost_flow</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="s2">"demand"</span><span class="p">)</span>
<span class="c1"># Build the flow into t_star</span>
<span class="k">for</span> <span class="n">source</span><span class="p">,</span> <span class="n">values</span> <span class="ow">in</span> <span class="n">flow_dict</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
<span class="k">for</span> <span class="n">target</span> <span class="ow">in</span> <span class="n">values</span><span class="p">:</span>
<span class="k">if</span> <span class="p">(</span><span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">t_star</span><span class="o">.</span><span class="n">edges</span> <span class="ow">and</span> <span class="n">values</span><span class="p">[</span><span class="n">target</span><span class="p">]</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
<span class="c1"># IF values[target] > 0 we have to add that many edges</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">values</span><span class="p">[</span><span class="n">target</span><span class="p">]):</span>
<span class="n">t_star</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
<span class="c1"># Return the shortcut eulerian circuit</span>
<span class="n">circuit</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">eulerian_circuit</span><span class="p">(</span><span class="n">t_star</span><span class="p">,</span> <span class="n">source</span><span class="o">=</span><span class="n">source</span><span class="p">)</span>
<span class="k">return</span> <span class="n">_shortcutting</span><span class="p">(</span><span class="n">circuit</span><span class="p">)</span></div>
<span class="k">def</span> <span class="nf">held_karp_ascent</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Minimizes the Held-Karp relaxation of the TSP for `G`</span>
<span class="sd"> Solves the Held-Karp relaxation of the input complete digraph and scales</span>
<span class="sd"> the output solution for use in the Asadpour [1]_ ASTP algorithm.</span>
<span class="sd"> The Held-Karp relaxation defines the lower bound for solutions to the</span>
<span class="sd"> ATSP, although it does return a fractional solution. This is used in the</span>
<span class="sd"> Asadpour algorithm as an initial solution which is later rounded to a</span>
<span class="sd"> integral tree within the spanning tree polytopes. This function solves</span>
<span class="sd"> the relaxation with the branch and bound method in [2]_.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : nx.DiGraph</span>
<span class="sd"> The graph should be a complete weighted directed graph.</span>
<span class="sd"> The distance between all paris of nodes should be included.</span>
<span class="sd"> weight : string, optional (default="weight")</span>
<span class="sd"> Edge data key corresponding to the edge weight.</span>
<span class="sd"> If any edge does not have this attribute the weight is set to 1.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> OPT : float</span>
<span class="sd"> The cost for the optimal solution to the Held-Karp relaxation</span>
<span class="sd"> z : dict or nx.Graph</span>
<span class="sd"> A symmetrized and scaled version of the optimal solution to the</span>
<span class="sd"> Held-Karp relaxation for use in the Asadpour algorithm.</span>
<span class="sd"> If an integral solution is found, then that is an optimal solution for</span>
<span class="sd"> the ATSP problem and that is returned instead.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] A. Asadpour, M. X. Goemans, A. Madry, S. O. Gharan, and A. Saberi,</span>
<span class="sd"> An o(log n/log log n)-approximation algorithm for the asymmetric</span>
<span class="sd"> traveling salesman problem, Operations research, 65 (2017),</span>
<span class="sd"> pp. 1043–1061</span>
<span class="sd"> .. [2] M. Held, R. M. Karp, The traveling-salesman problem and minimum</span>
<span class="sd"> spanning trees, Operations Research, 1970-11-01, Vol. 18 (6),</span>
<span class="sd"> pp.1138-1162</span>
<span class="sd"> """</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">scipy.optimize</span> <span class="k">as</span> <span class="nn">optimize</span>
<span class="k">def</span> <span class="nf">k_pi</span><span class="p">():</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Find the set of minimum 1-Arborescences for G at point pi.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> Set</span>
<span class="sd"> The set of minimum 1-Arborescences</span>
<span class="sd"> """</span>
<span class="c1"># Create a copy of G without vertex 1.</span>
<span class="n">G_1</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">minimum_1_arborescences</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
<span class="n">minimum_1_arborescence_weight</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">inf</span>
<span class="c1"># node is node '1' in the Held and Karp paper</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="fm">__iter__</span><span class="p">())</span>
<span class="n">G_1</span><span class="o">.</span><span class="n">remove_node</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="c1"># Iterate over the spanning arborescences of the graph until we know</span>
<span class="c1"># that we have found the minimum 1-arborescences. My proposed strategy</span>
<span class="c1"># is to find the most extensive root to connect to from 'node 1' and</span>
<span class="c1"># the least expensive one. We then iterate over arborescences until</span>
<span class="c1"># the cost of the basic arborescence is the cost of the minimum one</span>
<span class="c1"># plus the difference between the most and least expensive roots,</span>
<span class="c1"># that way the cost of connecting 'node 1' will by definition not by</span>
<span class="c1"># minimum</span>
<span class="n">min_root</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"node"</span><span class="p">:</span> <span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="p">:</span> <span class="n">math</span><span class="o">.</span><span class="n">inf</span><span class="p">}</span>
<span class="n">max_root</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"node"</span><span class="p">:</span> <span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="p">:</span> <span class="o">-</span><span class="n">math</span><span class="o">.</span><span class="n">inf</span><span class="p">}</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
<span class="k">if</span> <span class="n">d</span><span class="p">[</span><span class="n">weight</span><span class="p">]</span> <span class="o"><</span> <span class="n">min_root</span><span class="p">[</span><span class="n">weight</span><span class="p">]:</span>
<span class="n">min_root</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"node"</span><span class="p">:</span> <span class="n">v</span><span class="p">,</span> <span class="n">weight</span><span class="p">:</span> <span class="n">d</span><span class="p">[</span><span class="n">weight</span><span class="p">]}</span>
<span class="k">if</span> <span class="n">d</span><span class="p">[</span><span class="n">weight</span><span class="p">]</span> <span class="o">></span> <span class="n">max_root</span><span class="p">[</span><span class="n">weight</span><span class="p">]:</span>
<span class="n">max_root</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"node"</span><span class="p">:</span> <span class="n">v</span><span class="p">,</span> <span class="n">weight</span><span class="p">:</span> <span class="n">d</span><span class="p">[</span><span class="n">weight</span><span class="p">]}</span>
<span class="n">min_in_edge</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">in_edges</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="kc">True</span><span class="p">),</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="n">weight</span><span class="p">])</span>
<span class="n">min_root</span><span class="p">[</span><span class="n">weight</span><span class="p">]</span> <span class="o">=</span> <span class="n">min_root</span><span class="p">[</span><span class="n">weight</span><span class="p">]</span> <span class="o">+</span> <span class="n">min_in_edge</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="n">weight</span><span class="p">]</span>
<span class="n">max_root</span><span class="p">[</span><span class="n">weight</span><span class="p">]</span> <span class="o">=</span> <span class="n">max_root</span><span class="p">[</span><span class="n">weight</span><span class="p">]</span> <span class="o">+</span> <span class="n">min_in_edge</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="n">weight</span><span class="p">]</span>
<span class="n">min_arb_weight</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">inf</span>
<span class="k">for</span> <span class="n">arb</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">ArborescenceIterator</span><span class="p">(</span><span class="n">G_1</span><span class="p">):</span>
<span class="n">arb_weight</span> <span class="o">=</span> <span class="n">arb</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">weight</span><span class="p">)</span>
<span class="k">if</span> <span class="n">min_arb_weight</span> <span class="o">==</span> <span class="n">math</span><span class="o">.</span><span class="n">inf</span><span class="p">:</span>
<span class="n">min_arb_weight</span> <span class="o">=</span> <span class="n">arb_weight</span>
<span class="k">elif</span> <span class="n">arb_weight</span> <span class="o">></span> <span class="n">min_arb_weight</span> <span class="o">+</span> <span class="n">max_root</span><span class="p">[</span><span class="n">weight</span><span class="p">]</span> <span class="o">-</span> <span class="n">min_root</span><span class="p">[</span><span class="n">weight</span><span class="p">]:</span>
<span class="k">break</span>
<span class="c1"># We have to pick the root node of the arborescence for the out</span>
<span class="c1"># edge of the first vertex as that is the only node without an</span>
<span class="c1"># edge directed into it.</span>
<span class="k">for</span> <span class="n">N</span><span class="p">,</span> <span class="n">deg</span> <span class="ow">in</span> <span class="n">arb</span><span class="o">.</span><span class="n">in_degree</span><span class="p">:</span>
<span class="k">if</span> <span class="n">deg</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="c1"># root found</span>
<span class="n">arb</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="n">weight</span><span class="p">:</span> <span class="n">G</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">N</span><span class="p">][</span><span class="n">weight</span><span class="p">]})</span>
<span class="n">arb_weight</span> <span class="o">+=</span> <span class="n">G</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">N</span><span class="p">][</span><span class="n">weight</span><span class="p">]</span>
<span class="k">break</span>
<span class="c1"># We can pick the minimum weight in-edge for the vertex with</span>
<span class="c1"># a cycle. If there are multiple edges with the same, minimum</span>
<span class="c1"># weight, We need to add all of them.</span>
<span class="c1">#</span>
<span class="c1"># Delete the edge (N, v) so that we cannot pick it.</span>
<span class="n">edge_data</span> <span class="o">=</span> <span class="n">G</span><span class="p">[</span><span class="n">N</span><span class="p">][</span><span class="n">n</span><span class="p">]</span>
<span class="n">G</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="n">min_weight</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">in_edges</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">weight</span><span class="p">),</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])[</span><span class="mi">2</span><span class="p">]</span>
<span class="n">min_edges</span> <span class="o">=</span> <span class="p">[</span>
<span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span> <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">in_edges</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span> <span class="k">if</span> <span class="n">d</span> <span class="o">==</span> <span class="n">min_weight</span>
<span class="p">]</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">min_edges</span><span class="p">:</span>
<span class="n">new_arb</span> <span class="o">=</span> <span class="n">arb</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">new_arb</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="n">weight</span><span class="p">:</span> <span class="n">d</span><span class="p">})</span>
<span class="n">new_arb_weight</span> <span class="o">=</span> <span class="n">arb_weight</span> <span class="o">+</span> <span class="n">d</span>
<span class="c1"># Check to see the weight of the arborescence, if it is a</span>
<span class="c1"># new minimum, clear all of the old potential minimum</span>
<span class="c1"># 1-arborescences and add this is the only one. If its</span>
<span class="c1"># weight is above the known minimum, do not add it.</span>
<span class="k">if</span> <span class="n">new_arb_weight</span> <span class="o"><</span> <span class="n">minimum_1_arborescence_weight</span><span class="p">:</span>
<span class="n">minimum_1_arborescences</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span>
<span class="n">minimum_1_arborescence_weight</span> <span class="o">=</span> <span class="n">new_arb_weight</span>
<span class="c1"># We have a 1-arborescence, add it to the set</span>
<span class="k">if</span> <span class="n">new_arb_weight</span> <span class="o">==</span> <span class="n">minimum_1_arborescence_weight</span><span class="p">:</span>
<span class="n">minimum_1_arborescences</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">new_arb</span><span class="p">)</span>
<span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="o">**</span><span class="n">edge_data</span><span class="p">)</span>
<span class="k">return</span> <span class="n">minimum_1_arborescences</span>
<span class="k">def</span> <span class="nf">direction_of_ascent</span><span class="p">():</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Find the direction of ascent at point pi.</span>
<span class="sd"> See [1]_ for more information.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> dict</span>
<span class="sd"> A mapping from the nodes of the graph which represents the direction</span>
<span class="sd"> of ascent.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] M. Held, R. M. Karp, The traveling-salesman problem and minimum</span>
<span class="sd"> spanning trees, Operations Research, 1970-11-01, Vol. 18 (6),</span>
<span class="sd"> pp.1138-1162</span>
<span class="sd"> """</span>
<span class="c1"># 1. Set d equal to the zero n-vector.</span>
<span class="n">d</span> <span class="o">=</span> <span class="p">{}</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="n">d</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">del</span> <span class="n">n</span>
<span class="c1"># 2. Find a 1-Aborescence T^k such that k is in K(pi, d).</span>
<span class="n">minimum_1_arborescences</span> <span class="o">=</span> <span class="n">k_pi</span><span class="p">()</span>
<span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
<span class="c1"># Reduce K(pi) to K(pi, d)</span>
<span class="c1"># Find the arborescence in K(pi) which increases the lest in</span>
<span class="c1"># direction d</span>
<span class="n">min_k_d_weight</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">inf</span>
<span class="n">min_k_d</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">for</span> <span class="n">arborescence</span> <span class="ow">in</span> <span class="n">minimum_1_arborescences</span><span class="p">:</span>
<span class="n">weighted_cost</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">deg</span> <span class="ow">in</span> <span class="n">arborescence</span><span class="o">.</span><span class="n">degree</span><span class="p">:</span>
<span class="n">weighted_cost</span> <span class="o">+=</span> <span class="n">d</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">deg</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">if</span> <span class="n">weighted_cost</span> <span class="o"><</span> <span class="n">min_k_d_weight</span><span class="p">:</span>
<span class="n">min_k_d_weight</span> <span class="o">=</span> <span class="n">weighted_cost</span>
<span class="n">min_k_d</span> <span class="o">=</span> <span class="n">arborescence</span>
<span class="c1"># 3. If sum of d_i * v_{i, k} is greater than zero, terminate</span>
<span class="k">if</span> <span class="n">min_k_d_weight</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="n">d</span><span class="p">,</span> <span class="n">min_k_d</span>
<span class="c1"># 4. d_i = d_i + v_{i, k}</span>
<span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">deg</span> <span class="ow">in</span> <span class="n">min_k_d</span><span class="o">.</span><span class="n">degree</span><span class="p">:</span>
<span class="n">d</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">+=</span> <span class="n">deg</span> <span class="o">-</span> <span class="mi">2</span>
<span class="c1"># Check that we do not need to terminate because the direction</span>
<span class="c1"># of ascent does not exist. This is done with linear</span>
<span class="c1"># programming.</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">full</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">minimum_1_arborescences</span><span class="p">),</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)</span>
<span class="n">a_eq</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">minimum_1_arborescences</span><span class="p">)),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)</span>
<span class="n">b_eq</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)</span>
<span class="n">b_eq</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">for</span> <span class="n">arb_count</span><span class="p">,</span> <span class="n">arborescence</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">minimum_1_arborescences</span><span class="p">):</span>
<span class="n">n_count</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
<span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">deg</span> <span class="ow">in</span> <span class="n">arborescence</span><span class="o">.</span><span class="n">degree</span><span class="p">:</span>
<span class="n">a_eq</span><span class="p">[</span><span class="n">n_count</span><span class="p">][</span><span class="n">arb_count</span><span class="p">]</span> <span class="o">=</span> <span class="n">deg</span> <span class="o">-</span> <span class="mi">2</span>
<span class="n">n_count</span> <span class="o">-=</span> <span class="mi">1</span>
<span class="n">a_eq</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)][</span><span class="n">arb_count</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">program_result</span> <span class="o">=</span> <span class="n">optimize</span><span class="o">.</span><span class="n">linprog</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">A_eq</span><span class="o">=</span><span class="n">a_eq</span><span class="p">,</span> <span class="n">b_eq</span><span class="o">=</span><span class="n">b_eq</span><span class="p">)</span>
<span class="c1"># If the constants exist, then the direction of ascent doesn't</span>
<span class="k">if</span> <span class="n">program_result</span><span class="o">.</span><span class="n">success</span><span class="p">:</span>
<span class="c1"># There is no direction of ascent</span>
<span class="k">return</span> <span class="kc">None</span><span class="p">,</span> <span class="n">minimum_1_arborescences</span>
<span class="c1"># 5. GO TO 2</span>
<span class="k">def</span> <span class="nf">find_epsilon</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Given the direction of ascent at pi, find the maximum distance we can go</span>
<span class="sd"> in that direction.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> k_xy : set</span>
<span class="sd"> The set of 1-arborescences which have the minimum rate of increase</span>
<span class="sd"> in the direction of ascent</span>
<span class="sd"> d : dict</span>
<span class="sd"> The direction of ascent</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> float</span>
<span class="sd"> The distance we can travel in direction `d`</span>
<span class="sd"> """</span>
<span class="n">min_epsilon</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">inf</span>
<span class="k">for</span> <span class="n">e_u</span><span class="p">,</span> <span class="n">e_v</span><span class="p">,</span> <span class="n">e_w</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">weight</span><span class="p">):</span>
<span class="k">if</span> <span class="p">(</span><span class="n">e_u</span><span class="p">,</span> <span class="n">e_v</span><span class="p">)</span> <span class="ow">in</span> <span class="n">k</span><span class="o">.</span><span class="n">edges</span><span class="p">:</span>
<span class="k">continue</span>
<span class="c1"># Now, I have found a condition which MUST be true for the edges to</span>
<span class="c1"># be a valid substitute. The edge in the graph which is the</span>
<span class="c1"># substitute is the one with the same terminated end. This can be</span>
<span class="c1"># checked rather simply.</span>
<span class="c1">#</span>
<span class="c1"># Find the edge within k which is the substitute. Because k is a</span>
<span class="c1"># 1-arborescence, we know that they is only one such edges</span>
<span class="c1"># leading into every vertex.</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">k</span><span class="o">.</span><span class="n">in_edges</span><span class="p">(</span><span class="n">e_v</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">weight</span><span class="p">))</span> <span class="o">></span> <span class="mi">1</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">Exception</span>
<span class="n">sub_u</span><span class="p">,</span> <span class="n">sub_v</span><span class="p">,</span> <span class="n">sub_w</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">k</span><span class="o">.</span><span class="n">in_edges</span><span class="p">(</span><span class="n">e_v</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span><span class="o">.</span><span class="fm">__iter__</span><span class="p">())</span>
<span class="n">k</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">e_u</span><span class="p">,</span> <span class="n">e_v</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="n">weight</span><span class="p">:</span> <span class="n">e_w</span><span class="p">})</span>
<span class="n">k</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">sub_u</span><span class="p">,</span> <span class="n">sub_v</span><span class="p">)</span>
<span class="k">if</span> <span class="p">(</span>
<span class="nb">max</span><span class="p">(</span><span class="n">d</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">k</span><span class="o">.</span><span class="n">in_degree</span><span class="p">())</span> <span class="o"><=</span> <span class="mi">1</span>
<span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">==</span> <span class="n">k</span><span class="o">.</span><span class="n">number_of_edges</span><span class="p">()</span>
<span class="ow">and</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_weakly_connected</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
<span class="p">):</span>
<span class="c1"># Ascent method calculation</span>
<span class="k">if</span> <span class="n">d</span><span class="p">[</span><span class="n">sub_u</span><span class="p">]</span> <span class="o">==</span> <span class="n">d</span><span class="p">[</span><span class="n">e_u</span><span class="p">]</span> <span class="ow">or</span> <span class="n">sub_w</span> <span class="o">==</span> <span class="n">e_w</span><span class="p">:</span>
<span class="c1"># Revert to the original graph</span>
<span class="n">k</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">e_u</span><span class="p">,</span> <span class="n">e_v</span><span class="p">)</span>
<span class="n">k</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">sub_u</span><span class="p">,</span> <span class="n">sub_v</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="n">weight</span><span class="p">:</span> <span class="n">sub_w</span><span class="p">})</span>
<span class="k">continue</span>
<span class="n">epsilon</span> <span class="o">=</span> <span class="p">(</span><span class="n">sub_w</span> <span class="o">-</span> <span class="n">e_w</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">d</span><span class="p">[</span><span class="n">e_u</span><span class="p">]</span> <span class="o">-</span> <span class="n">d</span><span class="p">[</span><span class="n">sub_u</span><span class="p">])</span>
<span class="k">if</span> <span class="mi">0</span> <span class="o"><</span> <span class="n">epsilon</span> <span class="o"><</span> <span class="n">min_epsilon</span><span class="p">:</span>
<span class="n">min_epsilon</span> <span class="o">=</span> <span class="n">epsilon</span>
<span class="c1"># Revert to the original graph</span>
<span class="n">k</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">e_u</span><span class="p">,</span> <span class="n">e_v</span><span class="p">)</span>
<span class="n">k</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">sub_u</span><span class="p">,</span> <span class="n">sub_v</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="n">weight</span><span class="p">:</span> <span class="n">sub_w</span><span class="p">})</span>
<span class="k">return</span> <span class="n">min_epsilon</span>
<span class="c1"># I have to know that the elements in pi correspond to the correct elements</span>
<span class="c1"># in the direction of ascent, even if the node labels are not integers.</span>
<span class="c1"># Thus, I will use dictionaries to made that mapping.</span>
<span class="n">pi_dict</span> <span class="o">=</span> <span class="p">{}</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="n">pi_dict</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">del</span> <span class="n">n</span>
<span class="n">original_edge_weights</span> <span class="o">=</span> <span class="p">{}</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
<span class="n">original_edge_weights</span><span class="p">[(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)]</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="n">weight</span><span class="p">]</span>
<span class="n">dir_ascent</span><span class="p">,</span> <span class="n">k_d</span> <span class="o">=</span> <span class="n">direction_of_ascent</span><span class="p">()</span>
<span class="k">while</span> <span class="n">dir_ascent</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">max_distance</span> <span class="o">=</span> <span class="n">find_epsilon</span><span class="p">(</span><span class="n">k_d</span><span class="p">,</span> <span class="n">dir_ascent</span><span class="p">)</span>
<span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">dir_ascent</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
<span class="n">pi_dict</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">+=</span> <span class="n">max_distance</span> <span class="o">*</span> <span class="n">v</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
<span class="n">d</span><span class="p">[</span><span class="n">weight</span><span class="p">]</span> <span class="o">=</span> <span class="n">original_edge_weights</span><span class="p">[(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)]</span> <span class="o">+</span> <span class="n">pi_dict</span><span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="n">dir_ascent</span><span class="p">,</span> <span class="n">k_d</span> <span class="o">=</span> <span class="n">direction_of_ascent</span><span class="p">()</span>
<span class="c1"># k_d is no longer an individual 1-arborescence but rather a set of</span>
<span class="c1"># minimal 1-arborescences at the maximum point of the polytope and should</span>
<span class="c1"># be reflected as such</span>
<span class="n">k_max</span> <span class="o">=</span> <span class="n">k_d</span>
<span class="c1"># Search for a cycle within k_max. If a cycle exists, return it as the</span>
<span class="c1"># solution</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">k_max</span><span class="p">:</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">([</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">k</span> <span class="k">if</span> <span class="n">k</span><span class="o">.</span><span class="n">degree</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">])</span> <span class="o">==</span> <span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">():</span>
<span class="c1"># Tour found</span>
<span class="k">return</span> <span class="n">k</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">weight</span><span class="p">),</span> <span class="n">k</span>
<span class="c1"># Write the original edge weights back to G and every member of k_max at</span>
<span class="c1"># the maximum point. Also average the number of times that edge appears in</span>
<span class="c1"># the set of minimal 1-arborescences.</span>
<span class="n">x_star</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">size_k_max</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">k_max</span><span class="p">)</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
<span class="n">edge_count</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">d</span><span class="p">[</span><span class="n">weight</span><span class="p">]</span> <span class="o">=</span> <span class="n">original_edge_weights</span><span class="p">[(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)]</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">k_max</span><span class="p">:</span>
<span class="k">if</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="ow">in</span> <span class="n">k</span><span class="o">.</span><span class="n">edges</span><span class="p">():</span>
<span class="n">edge_count</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">k</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">][</span><span class="n">weight</span><span class="p">]</span> <span class="o">=</span> <span class="n">original_edge_weights</span><span class="p">[(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)]</span>
<span class="n">x_star</span><span class="p">[(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)]</span> <span class="o">=</span> <span class="n">edge_count</span> <span class="o">/</span> <span class="n">size_k_max</span>
<span class="c1"># Now symmetrize the edges in x_star and scale them according to (5) in</span>
<span class="c1"># reference [1]</span>
<span class="n">z_star</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">scale_factor</span> <span class="o">=</span> <span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">x_star</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
<span class="n">frequency</span> <span class="o">=</span> <span class="n">x_star</span><span class="p">[(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)]</span> <span class="o">+</span> <span class="n">x_star</span><span class="p">[(</span><span class="n">v</span><span class="p">,</span> <span class="n">u</span><span class="p">)]</span>
<span class="k">if</span> <span class="n">frequency</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
<span class="n">z_star</span><span class="p">[(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)]</span> <span class="o">=</span> <span class="n">scale_factor</span> <span class="o">*</span> <span class="n">frequency</span>
<span class="k">del</span> <span class="n">x_star</span>
<span class="c1"># Return the optimal weight and the z dict</span>
<span class="k">return</span> <span class="nb">next</span><span class="p">(</span><span class="n">k_max</span><span class="o">.</span><span class="fm">__iter__</span><span class="p">())</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">weight</span><span class="p">),</span> <span class="n">z_star</span>
<span class="k">def</span> <span class="nf">spanning_tree_distribution</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> Find the asadpour exponential distribution of spanning trees.</span>
<span class="sd"> Solves the Maximum Entropy Convex Program in the Asadpour algorithm [1]_</span>
<span class="sd"> using the approach in section 7 to build an exponential distribution of</span>
<span class="sd"> undirected spanning trees.</span>
<span class="sd"> This algorithm ensures that the probability of any edge in a spanning</span>
<span class="sd"> tree is proportional to the sum of the probabilities of the tress</span>
<span class="sd"> containing that edge over the sum of the probabilities of all spanning</span>
<span class="sd"> trees of the graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : nx.MultiGraph</span>
<span class="sd"> The undirected support graph for the Held Karp relaxation</span>
<span class="sd"> z : dict</span>
<span class="sd"> The output of `held_karp_ascent()`, a scaled version of the Held-Karp</span>
<span class="sd"> solution.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> gamma : dict</span>
<span class="sd"> The probability distribution which approximately preserves the marginal</span>
<span class="sd"> probabilities of `z`.</span>
<span class="sd"> """</span>
<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">exp</span>
<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">log</span> <span class="k">as</span> <span class="n">ln</span>
<span class="k">def</span> <span class="nf">q</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> The value of q(e) is described in the Asadpour paper is "the</span>
<span class="sd"> probability that edge e will be included in a spanning tree T that is</span>
<span class="sd"> chosen with probability proportional to exp(gamma(T))" which</span>
<span class="sd"> basically means that it is the total probability of the edge appearing</span>
<span class="sd"> across the whole distribution.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> e : tuple</span>
<span class="sd"> The `(u, v)` tuple describing the edge we are interested in</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> float</span>
<span class="sd"> The probability that a spanning tree chosen according to the</span>
<span class="sd"> current values of gamma will include edge `e`.</span>
<span class="sd"> """</span>
<span class="c1"># Create the laplacian matrices</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
<span class="n">d</span><span class="p">[</span><span class="n">lambda_key</span><span class="p">]</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="n">gamma</span><span class="p">[(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)])</span>
<span class="n">G_Kirchhoff</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">total_spanning_tree_weight</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">lambda_key</span><span class="p">)</span>
<span class="n">G_e</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">contracted_edge</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">self_loops</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="n">G_e_Kirchhoff</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">total_spanning_tree_weight</span><span class="p">(</span><span class="n">G_e</span><span class="p">,</span> <span class="n">lambda_key</span><span class="p">)</span>
<span class="c1"># Multiply by the weight of the contracted edge since it is not included</span>
<span class="c1"># in the total weight of the contracted graph.</span>
<span class="k">return</span> <span class="n">exp</span><span class="p">(</span><span class="n">gamma</span><span class="p">[(</span><span class="n">e</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">e</span><span class="p">[</span><span class="mi">1</span><span class="p">])])</span> <span class="o">*</span> <span class="n">G_e_Kirchhoff</span> <span class="o">/</span> <span class="n">G_Kirchhoff</span>
<span class="c1"># initialize gamma to the zero dict</span>
<span class="n">gamma</span> <span class="o">=</span> <span class="p">{}</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">:</span>
<span class="n">gamma</span><span class="p">[(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="c1"># set epsilon</span>
<span class="n">EPSILON</span> <span class="o">=</span> <span class="mf">0.2</span>
<span class="c1"># pick an edge attribute name that is unlikely to be in the graph</span>
<span class="n">lambda_key</span> <span class="o">=</span> <span class="s2">"spanning_tree_distribution's secret attribute name for lambda"</span>
<span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
<span class="c1"># We need to know that know that no values of q_e are greater than</span>
<span class="c1"># (1 + epsilon) * z_e, however changing one gamma value can increase the</span>
<span class="c1"># value of a different q_e, so we have to complete the for loop without</span>
<span class="c1"># changing anything for the condition to be meet</span>
<span class="n">in_range_count</span> <span class="o">=</span> <span class="mi">0</span>
<span class="c1"># Search for an edge with q_e > (1 + epsilon) * z_e</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">gamma</span><span class="p">:</span>
<span class="n">e</span> <span class="o">=</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="n">q_e</span> <span class="o">=</span> <span class="n">q</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
<span class="n">z_e</span> <span class="o">=</span> <span class="n">z</span><span class="p">[</span><span class="n">e</span><span class="p">]</span>
<span class="k">if</span> <span class="n">q_e</span> <span class="o">></span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">EPSILON</span><span class="p">)</span> <span class="o">*</span> <span class="n">z_e</span><span class="p">:</span>
<span class="n">delta</span> <span class="o">=</span> <span class="n">ln</span><span class="p">(</span>
<span class="p">(</span><span class="n">q_e</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">EPSILON</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">z_e</span><span class="p">))</span>
<span class="o">/</span> <span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">q_e</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">EPSILON</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">z_e</span><span class="p">)</span>
<span class="p">)</span>
<span class="n">gamma</span><span class="p">[</span><span class="n">e</span><span class="p">]</span> <span class="o">-=</span> <span class="n">delta</span>
<span class="c1"># Check that delta had the desired effect</span>
<span class="n">new_q_e</span> <span class="o">=</span> <span class="n">q</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
<span class="n">desired_q_e</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">EPSILON</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">z_e</span>
<span class="k">if</span> <span class="nb">round</span><span class="p">(</span><span class="n">new_q_e</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">round</span><span class="p">(</span><span class="n">desired_q_e</span><span class="p">,</span> <span class="mi">8</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span>
<span class="sa">f</span><span class="s2">"Unable to modify probability for edge (</span><span class="si">{</span><span class="n">u</span><span class="si">}</span><span class="s2">, </span><span class="si">{</span><span class="n">v</span><span class="si">}</span><span class="s2">)"</span>
<span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">in_range_count</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="c1"># Check if the for loop terminated without changing any gamma</span>
<span class="k">if</span> <span class="n">in_range_count</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">gamma</span><span class="p">):</span>
<span class="k">break</span>
<span class="c1"># Remove the new edge attributes</span>
<span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
<span class="k">if</span> <span class="n">lambda_key</span> <span class="ow">in</span> <span class="n">d</span><span class="p">:</span>
<span class="k">del</span> <span class="n">d</span><span class="p">[</span><span class="n">lambda_key</span><span class="p">]</span>
<span class="k">return</span> <span class="n">gamma</span>
<div class="viewcode-block" id="greedy_tsp"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.approximation.traveling_salesman.greedy_tsp.html#networkx.algorithms.approximation.traveling_salesman.greedy_tsp">[docs]</a><span class="k">def</span> <span class="nf">greedy_tsp</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">,</span> <span class="n">source</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Return a low cost cycle starting at `source` and its cost.</span>
<span class="sd"> This approximates a solution to the traveling salesman problem.</span>
<span class="sd"> It finds a cycle of all the nodes that a salesman can visit in order</span>
<span class="sd"> to visit many nodes while minimizing total distance.</span>
<span class="sd"> It uses a simple greedy algorithm.</span>
<span class="sd"> In essence, this function returns a large cycle given a source point</span>
<span class="sd"> for which the total cost of the cycle is minimized.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : Graph</span>
<span class="sd"> The Graph should be a complete weighted undirected graph.</span>
<span class="sd"> The distance between all pairs of nodes should be included.</span>
<span class="sd"> weight : string, optional (default="weight")</span>
<span class="sd"> Edge data key corresponding to the edge weight.</span>
<span class="sd"> If any edge does not have this attribute the weight is set to 1.</span>
<span class="sd"> source : node, optional (default: first node in list(G))</span>
<span class="sd"> Starting node. If None, defaults to ``next(iter(G))``</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> cycle : list of nodes</span>
<span class="sd"> Returns the cycle (list of nodes) that a salesman</span>
<span class="sd"> can follow to minimize total weight of the trip.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If `G` is not complete, the algorithm raises an exception.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms import approximation as approx</span>
<span class="sd"> >>> G = nx.DiGraph()</span>
<span class="sd"> >>> G.add_weighted_edges_from({</span>
<span class="sd"> ... ("A", "B", 3), ("A", "C", 17), ("A", "D", 14), ("B", "A", 3),</span>
<span class="sd"> ... ("B", "C", 12), ("B", "D", 16), ("C", "A", 13),("C", "B", 12),</span>
<span class="sd"> ... ("C", "D", 4), ("D", "A", 14), ("D", "B", 15), ("D", "C", 2)</span>
<span class="sd"> ... })</span>
<span class="sd"> >>> cycle = approx.greedy_tsp(G, source="D")</span>
<span class="sd"> >>> cost = sum(G[n][nbr]["weight"] for n, nbr in nx.utils.pairwise(cycle))</span>
<span class="sd"> >>> cycle</span>
<span class="sd"> ['D', 'C', 'B', 'A', 'D']</span>
<span class="sd"> >>> cost</span>
<span class="sd"> 31</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This implementation of a greedy algorithm is based on the following:</span>
<span class="sd"> - The algorithm adds a node to the solution at every iteration.</span>
<span class="sd"> - The algorithm selects a node not already in the cycle whose connection</span>
<span class="sd"> to the previous node adds the least cost to the cycle.</span>
<span class="sd"> A greedy algorithm does not always give the best solution.</span>
<span class="sd"> However, it can construct a first feasible solution which can</span>
<span class="sd"> be passed as a parameter to an iterative improvement algorithm such</span>
<span class="sd"> as Simulated Annealing, or Threshold Accepting.</span>
<span class="sd"> Time complexity: It has a running time $O(|V|^2)$</span>
<span class="sd"> """</span>
<span class="c1"># Check that G is a complete graph</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
<span class="c1"># This check ignores selfloops which is what we want here.</span>
<span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nbrdict</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">n</span> <span class="ow">in</span> <span class="n">nbrdict</span><span class="p">)</span> <span class="o">!=</span> <span class="n">N</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">nbrdict</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">adj</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"G must be a complete graph."</span><span class="p">)</span>
<span class="k">if</span> <span class="n">source</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">source</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">utils</span><span class="o">.</span><span class="n">arbitrary_element</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">number_of_nodes</span><span class="p">()</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
<span class="n">neighbor</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">neighbors</span><span class="p">(</span><span class="n">source</span><span class="p">))</span>
<span class="k">return</span> <span class="p">[</span><span class="n">source</span><span class="p">,</span> <span class="n">neighbor</span><span class="p">,</span> <span class="n">source</span><span class="p">]</span>
<span class="n">nodeset</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">nodeset</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">source</span><span class="p">)</span>
<span class="n">cycle</span> <span class="o">=</span> <span class="p">[</span><span class="n">source</span><span class="p">]</span>
<span class="n">next_node</span> <span class="o">=</span> <span class="n">source</span>
<span class="k">while</span> <span class="n">nodeset</span><span class="p">:</span>
<span class="n">nbrdict</span> <span class="o">=</span> <span class="n">G</span><span class="p">[</span><span class="n">next_node</span><span class="p">]</span>
<span class="n">next_node</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">nodeset</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">nbrdict</span><span class="p">[</span><span class="n">n</span><span class="p">]</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="n">cycle</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">next_node</span><span class="p">)</span>
<span class="n">nodeset</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">next_node</span><span class="p">)</span>
<span class="n">cycle</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cycle</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="k">return</span> <span class="n">cycle</span></div>
<div class="viewcode-block" id="simulated_annealing_tsp"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.approximation.traveling_salesman.simulated_annealing_tsp.html#networkx.algorithms.approximation.traveling_salesman.simulated_annealing_tsp">[docs]</a><span class="nd">@py_random_state</span><span class="p">(</span><span class="mi">9</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">simulated_annealing_tsp</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span>
<span class="n">init_cycle</span><span class="p">,</span>
<span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">,</span>
<span class="n">source</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">temp</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span>
<span class="n">move</span><span class="o">=</span><span class="s2">"1-1"</span><span class="p">,</span>
<span class="n">max_iterations</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
<span class="n">N_inner</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span>
<span class="n">alpha</span><span class="o">=</span><span class="mf">0.01</span><span class="p">,</span>
<span class="n">seed</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns an approximate solution to the traveling salesman problem.</span>
<span class="sd"> This function uses simulated annealing to approximate the minimal cost</span>
<span class="sd"> cycle through the nodes. Starting from a suboptimal solution, simulated</span>
<span class="sd"> annealing perturbs that solution, occasionally accepting changes that make</span>
<span class="sd"> the solution worse to escape from a locally optimal solution. The chance</span>
<span class="sd"> of accepting such changes decreases over the iterations to encourage</span>
<span class="sd"> an optimal result. In summary, the function returns a cycle starting</span>
<span class="sd"> at `source` for which the total cost is minimized. It also returns the cost.</span>
<span class="sd"> The chance of accepting a proposed change is related to a parameter called</span>
<span class="sd"> the temperature (annealing has a physical analogue of steel hardening</span>
<span class="sd"> as it cools). As the temperature is reduced, the chance of moves that</span>
<span class="sd"> increase cost goes down.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : Graph</span>
<span class="sd"> `G` should be a complete weighted undirected graph.</span>
<span class="sd"> The distance between all pairs of nodes should be included.</span>
<span class="sd"> init_cycle : list of all nodes or "greedy"</span>
<span class="sd"> The initial solution (a cycle through all nodes returning to the start).</span>
<span class="sd"> This argument has no default to make you think about it.</span>
<span class="sd"> If "greedy", use `greedy_tsp(G, weight)`.</span>
<span class="sd"> Other common starting cycles are `list(G) + [next(iter(G))]` or the final</span>
<span class="sd"> result of `simulated_annealing_tsp` when doing `threshold_accepting_tsp`.</span>
<span class="sd"> weight : string, optional (default="weight")</span>
<span class="sd"> Edge data key corresponding to the edge weight.</span>
<span class="sd"> If any edge does not have this attribute the weight is set to 1.</span>
<span class="sd"> source : node, optional (default: first node in list(G))</span>
<span class="sd"> Starting node. If None, defaults to ``next(iter(G))``</span>
<span class="sd"> temp : int, optional (default=100)</span>
<span class="sd"> The algorithm's temperature parameter. It represents the initial</span>
<span class="sd"> value of temperature</span>
<span class="sd"> move : "1-1" or "1-0" or function, optional (default="1-1")</span>
<span class="sd"> Indicator of what move to use when finding new trial solutions.</span>
<span class="sd"> Strings indicate two special built-in moves:</span>
<span class="sd"> - "1-1": 1-1 exchange which transposes the position</span>
<span class="sd"> of two elements of the current solution.</span>
<span class="sd"> The function called is :func:`swap_two_nodes`.</span>
<span class="sd"> For example if we apply 1-1 exchange in the solution</span>
<span class="sd"> ``A = [3, 2, 1, 4, 3]``</span>
<span class="sd"> we can get the following by the transposition of 1 and 4 elements:</span>
<span class="sd"> ``A' = [3, 2, 4, 1, 3]``</span>
<span class="sd"> - "1-0": 1-0 exchange which moves an node in the solution</span>
<span class="sd"> to a new position.</span>
<span class="sd"> The function called is :func:`move_one_node`.</span>
<span class="sd"> For example if we apply 1-0 exchange in the solution</span>
<span class="sd"> ``A = [3, 2, 1, 4, 3]``</span>
<span class="sd"> we can transfer the fourth element to the second position:</span>
<span class="sd"> ``A' = [3, 4, 2, 1, 3]``</span>
<span class="sd"> You may provide your own functions to enact a move from</span>
<span class="sd"> one solution to a neighbor solution. The function must take</span>
<span class="sd"> the solution as input along with a `seed` input to control</span>
<span class="sd"> random number generation (see the `seed` input here).</span>
<span class="sd"> Your function should maintain the solution as a cycle with</span>
<span class="sd"> equal first and last node and all others appearing once.</span>
<span class="sd"> Your function should return the new solution.</span>
<span class="sd"> max_iterations : int, optional (default=10)</span>
<span class="sd"> Declared done when this number of consecutive iterations of</span>
<span class="sd"> the outer loop occurs without any change in the best cost solution.</span>
<span class="sd"> N_inner : int, optional (default=100)</span>
<span class="sd"> The number of iterations of the inner loop.</span>
<span class="sd"> alpha : float between (0, 1), optional (default=0.01)</span>
<span class="sd"> Percentage of temperature decrease in each iteration</span>
<span class="sd"> of outer loop</span>
<span class="sd"> seed : integer, random_state, or None (default)</span>
<span class="sd"> Indicator of random number generation state.</span>
<span class="sd"> See :ref:`Randomness<randomness>`.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> cycle : list of nodes</span>
<span class="sd"> Returns the cycle (list of nodes) that a salesman</span>
<span class="sd"> can follow to minimize total weight of the trip.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If `G` is not complete the algorithm raises an exception.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms import approximation as approx</span>
<span class="sd"> >>> G = nx.DiGraph()</span>
<span class="sd"> >>> G.add_weighted_edges_from({</span>
<span class="sd"> ... ("A", "B", 3), ("A", "C", 17), ("A", "D", 14), ("B", "A", 3),</span>
<span class="sd"> ... ("B", "C", 12), ("B", "D", 16), ("C", "A", 13),("C", "B", 12),</span>
<span class="sd"> ... ("C", "D", 4), ("D", "A", 14), ("D", "B", 15), ("D", "C", 2)</span>
<span class="sd"> ... })</span>
<span class="sd"> >>> cycle = approx.simulated_annealing_tsp(G, "greedy", source="D")</span>
<span class="sd"> >>> cost = sum(G[n][nbr]["weight"] for n, nbr in nx.utils.pairwise(cycle))</span>
<span class="sd"> >>> cycle</span>
<span class="sd"> ['D', 'C', 'B', 'A', 'D']</span>
<span class="sd"> >>> cost</span>
<span class="sd"> 31</span>
<span class="sd"> >>> incycle = ["D", "B", "A", "C", "D"]</span>
<span class="sd"> >>> cycle = approx.simulated_annealing_tsp(G, incycle, source="D")</span>
<span class="sd"> >>> cost = sum(G[n][nbr]["weight"] for n, nbr in nx.utils.pairwise(cycle))</span>
<span class="sd"> >>> cycle</span>
<span class="sd"> ['D', 'C', 'B', 'A', 'D']</span>
<span class="sd"> >>> cost</span>
<span class="sd"> 31</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Simulated Annealing is a metaheuristic local search algorithm.</span>
<span class="sd"> The main characteristic of this algorithm is that it accepts</span>
<span class="sd"> even solutions which lead to the increase of the cost in order</span>
<span class="sd"> to escape from low quality local optimal solutions.</span>
<span class="sd"> This algorithm needs an initial solution. If not provided, it is</span>
<span class="sd"> constructed by a simple greedy algorithm. At every iteration, the</span>
<span class="sd"> algorithm selects thoughtfully a neighbor solution.</span>
<span class="sd"> Consider $c(x)$ cost of current solution and $c(x')$ cost of a</span>
<span class="sd"> neighbor solution.</span>
<span class="sd"> If $c(x') - c(x) <= 0$ then the neighbor solution becomes the current</span>
<span class="sd"> solution for the next iteration. Otherwise, the algorithm accepts</span>
<span class="sd"> the neighbor solution with probability $p = exp - ([c(x') - c(x)] / temp)$.</span>
<span class="sd"> Otherwise the current solution is retained.</span>
<span class="sd"> `temp` is a parameter of the algorithm and represents temperature.</span>
<span class="sd"> Time complexity:</span>
<span class="sd"> For $N_i$ iterations of the inner loop and $N_o$ iterations of the</span>
<span class="sd"> outer loop, this algorithm has running time $O(N_i * N_o * |V|)$.</span>
<span class="sd"> For more information and how the algorithm is inspired see:</span>
<span class="sd"> http://en.wikipedia.org/wiki/Simulated_annealing</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">move</span> <span class="o">==</span> <span class="s2">"1-1"</span><span class="p">:</span>
<span class="n">move</span> <span class="o">=</span> <span class="n">swap_two_nodes</span>
<span class="k">elif</span> <span class="n">move</span> <span class="o">==</span> <span class="s2">"1-0"</span><span class="p">:</span>
<span class="n">move</span> <span class="o">=</span> <span class="n">move_one_node</span>
<span class="k">if</span> <span class="n">init_cycle</span> <span class="o">==</span> <span class="s2">"greedy"</span><span class="p">:</span>
<span class="c1"># Construct an initial solution using a greedy algorithm.</span>
<span class="n">cycle</span> <span class="o">=</span> <span class="n">greedy_tsp</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">,</span> <span class="n">source</span><span class="o">=</span><span class="n">source</span><span class="p">)</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">number_of_nodes</span><span class="p">()</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
<span class="k">return</span> <span class="n">cycle</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">cycle</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">init_cycle</span><span class="p">)</span>
<span class="k">if</span> <span class="n">source</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">source</span> <span class="o">=</span> <span class="n">cycle</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">elif</span> <span class="n">source</span> <span class="o">!=</span> <span class="n">cycle</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"source must be first node in init_cycle"</span><span class="p">)</span>
<span class="k">if</span> <span class="n">cycle</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">cycle</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"init_cycle must be a cycle. (return to start)"</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">cycle</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">nbunch_iter</span><span class="p">(</span><span class="n">cycle</span><span class="p">)))</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"init_cycle should be a cycle over all nodes in G."</span><span class="p">)</span>
<span class="c1"># Check that G is a complete graph</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
<span class="c1"># This check ignores selfloops which is what we want here.</span>
<span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nbrdict</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">n</span> <span class="ow">in</span> <span class="n">nbrdict</span><span class="p">)</span> <span class="o">!=</span> <span class="n">N</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">nbrdict</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">adj</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"G must be a complete graph."</span><span class="p">)</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">number_of_nodes</span><span class="p">()</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
<span class="n">neighbor</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">neighbors</span><span class="p">(</span><span class="n">source</span><span class="p">))</span>
<span class="k">return</span> <span class="p">[</span><span class="n">source</span><span class="p">,</span> <span class="n">neighbor</span><span class="p">,</span> <span class="n">source</span><span class="p">]</span>
<span class="c1"># Find the cost of initial solution</span>
<span class="n">cost</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">pairwise</span><span class="p">(</span><span class="n">cycle</span><span class="p">))</span>
<span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">best_cycle</span> <span class="o">=</span> <span class="n">cycle</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">best_cost</span> <span class="o">=</span> <span class="n">cost</span>
<span class="k">while</span> <span class="n">count</span> <span class="o"><=</span> <span class="n">max_iterations</span> <span class="ow">and</span> <span class="n">temp</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
<span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_inner</span><span class="p">):</span>
<span class="n">adj_sol</span> <span class="o">=</span> <span class="n">move</span><span class="p">(</span><span class="n">cycle</span><span class="p">,</span> <span class="n">seed</span><span class="p">)</span>
<span class="n">adj_cost</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">pairwise</span><span class="p">(</span><span class="n">adj_sol</span><span class="p">))</span>
<span class="n">delta</span> <span class="o">=</span> <span class="n">adj_cost</span> <span class="o">-</span> <span class="n">cost</span>
<span class="k">if</span> <span class="n">delta</span> <span class="o"><=</span> <span class="mi">0</span><span class="p">:</span>
<span class="c1"># Set current solution the adjacent solution.</span>
<span class="n">cycle</span> <span class="o">=</span> <span class="n">adj_sol</span>
<span class="n">cost</span> <span class="o">=</span> <span class="n">adj_cost</span>
<span class="k">if</span> <span class="n">cost</span> <span class="o"><</span> <span class="n">best_cost</span><span class="p">:</span>
<span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">best_cycle</span> <span class="o">=</span> <span class="n">cycle</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">best_cost</span> <span class="o">=</span> <span class="n">cost</span>
<span class="k">else</span><span class="p">:</span>
<span class="c1"># Accept even a worse solution with probability p.</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">delta</span> <span class="o">/</span> <span class="n">temp</span><span class="p">)</span>
<span class="k">if</span> <span class="n">p</span> <span class="o">>=</span> <span class="n">seed</span><span class="o">.</span><span class="n">random</span><span class="p">():</span>
<span class="n">cycle</span> <span class="o">=</span> <span class="n">adj_sol</span>
<span class="n">cost</span> <span class="o">=</span> <span class="n">adj_cost</span>
<span class="n">temp</span> <span class="o">-=</span> <span class="n">temp</span> <span class="o">*</span> <span class="n">alpha</span>
<span class="k">return</span> <span class="n">best_cycle</span></div>
<div class="viewcode-block" id="threshold_accepting_tsp"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.approximation.traveling_salesman.threshold_accepting_tsp.html#networkx.algorithms.approximation.traveling_salesman.threshold_accepting_tsp">[docs]</a><span class="nd">@py_random_state</span><span class="p">(</span><span class="mi">9</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">threshold_accepting_tsp</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span>
<span class="n">init_cycle</span><span class="p">,</span>
<span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">,</span>
<span class="n">source</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">threshold</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
<span class="n">move</span><span class="o">=</span><span class="s2">"1-1"</span><span class="p">,</span>
<span class="n">max_iterations</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
<span class="n">N_inner</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span>
<span class="n">alpha</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span>
<span class="n">seed</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns an approximate solution to the traveling salesman problem.</span>
<span class="sd"> This function uses threshold accepting methods to approximate the minimal cost</span>
<span class="sd"> cycle through the nodes. Starting from a suboptimal solution, threshold</span>
<span class="sd"> accepting methods perturb that solution, accepting any changes that make</span>
<span class="sd"> the solution no worse than increasing by a threshold amount. Improvements</span>
<span class="sd"> in cost are accepted, but so are changes leading to small increases in cost.</span>
<span class="sd"> This allows the solution to leave suboptimal local minima in solution space.</span>
<span class="sd"> The threshold is decreased slowly as iterations proceed helping to ensure</span>
<span class="sd"> an optimum. In summary, the function returns a cycle starting at `source`</span>
<span class="sd"> for which the total cost is minimized.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : Graph</span>
<span class="sd"> `G` should be a complete weighted undirected graph.</span>
<span class="sd"> The distance between all pairs of nodes should be included.</span>
<span class="sd"> init_cycle : list or "greedy"</span>
<span class="sd"> The initial solution (a cycle through all nodes returning to the start).</span>
<span class="sd"> This argument has no default to make you think about it.</span>
<span class="sd"> If "greedy", use `greedy_tsp(G, weight)`.</span>
<span class="sd"> Other common starting cycles are `list(G) + [next(iter(G))]` or the final</span>
<span class="sd"> result of `simulated_annealing_tsp` when doing `threshold_accepting_tsp`.</span>
<span class="sd"> weight : string, optional (default="weight")</span>
<span class="sd"> Edge data key corresponding to the edge weight.</span>
<span class="sd"> If any edge does not have this attribute the weight is set to 1.</span>
<span class="sd"> source : node, optional (default: first node in list(G))</span>
<span class="sd"> Starting node. If None, defaults to ``next(iter(G))``</span>
<span class="sd"> threshold : int, optional (default=1)</span>
<span class="sd"> The algorithm's threshold parameter. It represents the initial</span>
<span class="sd"> threshold's value</span>
<span class="sd"> move : "1-1" or "1-0" or function, optional (default="1-1")</span>
<span class="sd"> Indicator of what move to use when finding new trial solutions.</span>
<span class="sd"> Strings indicate two special built-in moves:</span>
<span class="sd"> - "1-1": 1-1 exchange which transposes the position</span>
<span class="sd"> of two elements of the current solution.</span>
<span class="sd"> The function called is :func:`swap_two_nodes`.</span>
<span class="sd"> For example if we apply 1-1 exchange in the solution</span>
<span class="sd"> ``A = [3, 2, 1, 4, 3]``</span>
<span class="sd"> we can get the following by the transposition of 1 and 4 elements:</span>
<span class="sd"> ``A' = [3, 2, 4, 1, 3]``</span>
<span class="sd"> - "1-0": 1-0 exchange which moves an node in the solution</span>
<span class="sd"> to a new position.</span>
<span class="sd"> The function called is :func:`move_one_node`.</span>
<span class="sd"> For example if we apply 1-0 exchange in the solution</span>
<span class="sd"> ``A = [3, 2, 1, 4, 3]``</span>
<span class="sd"> we can transfer the fourth element to the second position:</span>
<span class="sd"> ``A' = [3, 4, 2, 1, 3]``</span>
<span class="sd"> You may provide your own functions to enact a move from</span>
<span class="sd"> one solution to a neighbor solution. The function must take</span>
<span class="sd"> the solution as input along with a `seed` input to control</span>
<span class="sd"> random number generation (see the `seed` input here).</span>
<span class="sd"> Your function should maintain the solution as a cycle with</span>
<span class="sd"> equal first and last node and all others appearing once.</span>
<span class="sd"> Your function should return the new solution.</span>
<span class="sd"> max_iterations : int, optional (default=10)</span>
<span class="sd"> Declared done when this number of consecutive iterations of</span>
<span class="sd"> the outer loop occurs without any change in the best cost solution.</span>
<span class="sd"> N_inner : int, optional (default=100)</span>
<span class="sd"> The number of iterations of the inner loop.</span>
<span class="sd"> alpha : float between (0, 1), optional (default=0.1)</span>
<span class="sd"> Percentage of threshold decrease when there is at</span>
<span class="sd"> least one acceptance of a neighbor solution.</span>
<span class="sd"> If no inner loop moves are accepted the threshold remains unchanged.</span>
<span class="sd"> seed : integer, random_state, or None (default)</span>
<span class="sd"> Indicator of random number generation state.</span>
<span class="sd"> See :ref:`Randomness<randomness>`.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> cycle : list of nodes</span>
<span class="sd"> Returns the cycle (list of nodes) that a salesman</span>
<span class="sd"> can follow to minimize total weight of the trip.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If `G` is not complete the algorithm raises an exception.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms import approximation as approx</span>
<span class="sd"> >>> G = nx.DiGraph()</span>
<span class="sd"> >>> G.add_weighted_edges_from({</span>
<span class="sd"> ... ("A", "B", 3), ("A", "C", 17), ("A", "D", 14), ("B", "A", 3),</span>
<span class="sd"> ... ("B", "C", 12), ("B", "D", 16), ("C", "A", 13),("C", "B", 12),</span>
<span class="sd"> ... ("C", "D", 4), ("D", "A", 14), ("D", "B", 15), ("D", "C", 2)</span>
<span class="sd"> ... })</span>
<span class="sd"> >>> cycle = approx.threshold_accepting_tsp(G, "greedy", source="D")</span>
<span class="sd"> >>> cost = sum(G[n][nbr]["weight"] for n, nbr in nx.utils.pairwise(cycle))</span>
<span class="sd"> >>> cycle</span>
<span class="sd"> ['D', 'C', 'B', 'A', 'D']</span>
<span class="sd"> >>> cost</span>
<span class="sd"> 31</span>
<span class="sd"> >>> incycle = ["D", "B", "A", "C", "D"]</span>
<span class="sd"> >>> cycle = approx.threshold_accepting_tsp(G, incycle, source="D")</span>
<span class="sd"> >>> cost = sum(G[n][nbr]["weight"] for n, nbr in nx.utils.pairwise(cycle))</span>
<span class="sd"> >>> cycle</span>
<span class="sd"> ['D', 'C', 'B', 'A', 'D']</span>
<span class="sd"> >>> cost</span>
<span class="sd"> 31</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Threshold Accepting is a metaheuristic local search algorithm.</span>
<span class="sd"> The main characteristic of this algorithm is that it accepts</span>
<span class="sd"> even solutions which lead to the increase of the cost in order</span>
<span class="sd"> to escape from low quality local optimal solutions.</span>
<span class="sd"> This algorithm needs an initial solution. This solution can be</span>
<span class="sd"> constructed by a simple greedy algorithm. At every iteration, it</span>
<span class="sd"> selects thoughtfully a neighbor solution.</span>
<span class="sd"> Consider $c(x)$ cost of current solution and $c(x')$ cost of</span>
<span class="sd"> neighbor solution.</span>
<span class="sd"> If $c(x') - c(x) <= threshold$ then the neighbor solution becomes the current</span>
<span class="sd"> solution for the next iteration, where the threshold is named threshold.</span>
<span class="sd"> In comparison to the Simulated Annealing algorithm, the Threshold</span>
<span class="sd"> Accepting algorithm does not accept very low quality solutions</span>
<span class="sd"> (due to the presence of the threshold value). In the case of</span>
<span class="sd"> Simulated Annealing, even a very low quality solution can</span>
<span class="sd"> be accepted with probability $p$.</span>
<span class="sd"> Time complexity:</span>
<span class="sd"> It has a running time $O(m * n * |V|)$ where $m$ and $n$ are the number</span>
<span class="sd"> of times the outer and inner loop run respectively.</span>
<span class="sd"> For more information and how algorithm is inspired see:</span>
<span class="sd"> https://doi.org/10.1016/0021-9991(90)90201-B</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> simulated_annealing_tsp</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">move</span> <span class="o">==</span> <span class="s2">"1-1"</span><span class="p">:</span>
<span class="n">move</span> <span class="o">=</span> <span class="n">swap_two_nodes</span>
<span class="k">elif</span> <span class="n">move</span> <span class="o">==</span> <span class="s2">"1-0"</span><span class="p">:</span>
<span class="n">move</span> <span class="o">=</span> <span class="n">move_one_node</span>
<span class="k">if</span> <span class="n">init_cycle</span> <span class="o">==</span> <span class="s2">"greedy"</span><span class="p">:</span>
<span class="c1"># Construct an initial solution using a greedy algorithm.</span>
<span class="n">cycle</span> <span class="o">=</span> <span class="n">greedy_tsp</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">,</span> <span class="n">source</span><span class="o">=</span><span class="n">source</span><span class="p">)</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">number_of_nodes</span><span class="p">()</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
<span class="k">return</span> <span class="n">cycle</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">cycle</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">init_cycle</span><span class="p">)</span>
<span class="k">if</span> <span class="n">source</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">source</span> <span class="o">=</span> <span class="n">cycle</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">elif</span> <span class="n">source</span> <span class="o">!=</span> <span class="n">cycle</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"source must be first node in init_cycle"</span><span class="p">)</span>
<span class="k">if</span> <span class="n">cycle</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">cycle</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"init_cycle must be a cycle. (return to start)"</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">cycle</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">nbunch_iter</span><span class="p">(</span><span class="n">cycle</span><span class="p">)))</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"init_cycle is not all and only nodes."</span><span class="p">)</span>
<span class="c1"># Check that G is a complete graph</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
<span class="c1"># This check ignores selfloops which is what we want here.</span>
<span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nbrdict</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">n</span> <span class="ow">in</span> <span class="n">nbrdict</span><span class="p">)</span> <span class="o">!=</span> <span class="n">N</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">nbrdict</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">adj</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"G must be a complete graph."</span><span class="p">)</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">number_of_nodes</span><span class="p">()</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
<span class="n">neighbor</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">neighbors</span><span class="p">(</span><span class="n">source</span><span class="p">))[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">return</span> <span class="p">[</span><span class="n">source</span><span class="p">,</span> <span class="n">neighbor</span><span class="p">,</span> <span class="n">source</span><span class="p">]</span>
<span class="c1"># Find the cost of initial solution</span>
<span class="n">cost</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">pairwise</span><span class="p">(</span><span class="n">cycle</span><span class="p">))</span>
<span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">best_cycle</span> <span class="o">=</span> <span class="n">cycle</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">best_cost</span> <span class="o">=</span> <span class="n">cost</span>
<span class="k">while</span> <span class="n">count</span> <span class="o"><=</span> <span class="n">max_iterations</span><span class="p">:</span>
<span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">accepted</span> <span class="o">=</span> <span class="kc">False</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_inner</span><span class="p">):</span>
<span class="n">adj_sol</span> <span class="o">=</span> <span class="n">move</span><span class="p">(</span><span class="n">cycle</span><span class="p">,</span> <span class="n">seed</span><span class="p">)</span>
<span class="n">adj_cost</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">pairwise</span><span class="p">(</span><span class="n">adj_sol</span><span class="p">))</span>
<span class="n">delta</span> <span class="o">=</span> <span class="n">adj_cost</span> <span class="o">-</span> <span class="n">cost</span>
<span class="k">if</span> <span class="n">delta</span> <span class="o"><=</span> <span class="n">threshold</span><span class="p">:</span>
<span class="n">accepted</span> <span class="o">=</span> <span class="kc">True</span>
<span class="c1"># Set current solution the adjacent solution.</span>
<span class="n">cycle</span> <span class="o">=</span> <span class="n">adj_sol</span>
<span class="n">cost</span> <span class="o">=</span> <span class="n">adj_cost</span>
<span class="k">if</span> <span class="n">cost</span> <span class="o"><</span> <span class="n">best_cost</span><span class="p">:</span>
<span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">best_cycle</span> <span class="o">=</span> <span class="n">cycle</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">best_cost</span> <span class="o">=</span> <span class="n">cost</span>
<span class="k">if</span> <span class="n">accepted</span><span class="p">:</span>
<span class="n">threshold</span> <span class="o">-=</span> <span class="n">threshold</span> <span class="o">*</span> <span class="n">alpha</span>
<span class="k">return</span> <span class="n">best_cycle</span></div>
</pre></div>
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