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<a href="christofides_8h.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno">    1</span>&#160;<span class="comment">// Copyright 2010-2018 Google LLC</span></div>
<div class="line"><a name="l00002"></a><span class="lineno">    2</span>&#160;<span class="comment">// Licensed under the Apache License, Version 2.0 (the &quot;License&quot;);</span></div>
<div class="line"><a name="l00003"></a><span class="lineno">    3</span>&#160;<span class="comment">// you may not use this file except in compliance with the License.</span></div>
<div class="line"><a name="l00004"></a><span class="lineno">    4</span>&#160;<span class="comment">// You may obtain a copy of the License at</span></div>
<div class="line"><a name="l00005"></a><span class="lineno">    5</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00006"></a><span class="lineno">    6</span>&#160;<span class="comment">//     http://www.apache.org/licenses/LICENSE-2.0</span></div>
<div class="line"><a name="l00007"></a><span class="lineno">    7</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00008"></a><span class="lineno">    8</span>&#160;<span class="comment">// Unless required by applicable law or agreed to in writing, software</span></div>
<div class="line"><a name="l00009"></a><span class="lineno">    9</span>&#160;<span class="comment">// distributed under the License is distributed on an &quot;AS IS&quot; BASIS,</span></div>
<div class="line"><a name="l00010"></a><span class="lineno">   10</span>&#160;<span class="comment">// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.</span></div>
<div class="line"><a name="l00011"></a><span class="lineno">   11</span>&#160;<span class="comment">// See the License for the specific language governing permissions and</span></div>
<div class="line"><a name="l00012"></a><span class="lineno">   12</span>&#160;<span class="comment">// limitations under the License.</span></div>
<div class="line"><a name="l00013"></a><span class="lineno">   13</span>&#160; </div>
<div class="line"><a name="l00014"></a><span class="lineno">   14</span>&#160;<span class="comment">// ChristofidesPathSolver computes an approximate solution to the Traveling</span></div>
<div class="line"><a name="l00015"></a><span class="lineno">   15</span>&#160;<span class="comment">// Salesman Problen using the Christofides algorithm (c.f.</span></div>
<div class="line"><a name="l00016"></a><span class="lineno">   16</span>&#160;<span class="comment">// https://en.wikipedia.org/wiki/Christofides_algorithm).</span></div>
<div class="line"><a name="l00017"></a><span class="lineno">   17</span>&#160;<span class="comment">// Note that the algorithm guarantees finding a solution within 3/2 of the</span></div>
<div class="line"><a name="l00018"></a><span class="lineno">   18</span>&#160;<span class="comment">// optimum. Its complexity is O(n^2 * log(n)) where n is the number of nodes.</span></div>
<div class="line"><a name="l00019"></a><span class="lineno">   19</span>&#160; </div>
<div class="line"><a name="l00020"></a><span class="lineno">   20</span>&#160;<span class="preprocessor">#ifndef OR_TOOLS_GRAPH_CHRISTOFIDES_H_</span></div>
<div class="line"><a name="l00021"></a><span class="lineno">   21</span>&#160;<span class="preprocessor">#define OR_TOOLS_GRAPH_CHRISTOFIDES_H_</span></div>
<div class="line"><a name="l00022"></a><span class="lineno">   22</span>&#160; </div>
<div class="line"><a name="l00023"></a><span class="lineno">   23</span>&#160;<span class="preprocessor">#include &quot;ortools/base/integral_types.h&quot;</span></div>
<div class="line"><a name="l00024"></a><span class="lineno">   24</span>&#160;<span class="preprocessor">#include &quot;ortools/base/logging.h&quot;</span></div>
<div class="line"><a name="l00025"></a><span class="lineno">   25</span>&#160;<span class="preprocessor">#include &quot;<a class="code" href="eulerian__path_8h.html">ortools/graph/eulerian_path.h</a>&quot;</span></div>
<div class="line"><a name="l00026"></a><span class="lineno">   26</span>&#160;<span class="preprocessor">#include &quot;<a class="code" href="graph_8h.html">ortools/graph/graph.h</a>&quot;</span></div>
<div class="line"><a name="l00027"></a><span class="lineno">   27</span>&#160;<span class="preprocessor">#include &quot;<a class="code" href="minimum__spanning__tree_8h.html">ortools/graph/minimum_spanning_tree.h</a>&quot;</span></div>
<div class="line"><a name="l00028"></a><span class="lineno">   28</span>&#160;<span class="preprocessor">#include &quot;ortools/graph/perfect_matching.h&quot;</span></div>
<div class="line"><a name="l00029"></a><span class="lineno">   29</span>&#160;<span class="preprocessor">#include &quot;ortools/linear_solver/linear_solver.h&quot;</span></div>
<div class="line"><a name="l00030"></a><span class="lineno">   30</span>&#160;<span class="preprocessor">#include &quot;ortools/linear_solver/linear_solver.pb.h&quot;</span></div>
<div class="line"><a name="l00031"></a><span class="lineno">   31</span>&#160;<span class="preprocessor">#include &quot;ortools/util/saturated_arithmetic.h&quot;</span></div>
<div class="line"><a name="l00032"></a><span class="lineno">   32</span>&#160; </div>
<div class="line"><a name="l00033"></a><span class="lineno"><a class="line" href="namespaceoperations__research.html">   33</a></span>&#160;<span class="keyword">namespace </span><a class="code" href="namespaceoperations__research.html">operations_research</a> {</div>
<div class="line"><a name="l00034"></a><span class="lineno">   34</span>&#160; </div>
<div class="line"><a name="l00035"></a><span class="lineno">   35</span>&#160;using ::util::CompleteGraph;</div>
<div class="line"><a name="l00036"></a><span class="lineno">   36</span>&#160; </div>
<div class="line"><a name="l00037"></a><span class="lineno">   37</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> CostType, <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> = int64,</div>
<div class="line"><a name="l00038"></a><span class="lineno">   38</span>&#160;          <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> = int32,</div>
<div class="line"><a name="l00039"></a><span class="lineno">   39</span>&#160;          <span class="keyword">typename</span> CostFunction = std::function&lt;CostType(<a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a>, <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a>)&gt;&gt;</div>
<div class="line"><a name="l00040"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_christofides_path_solver.html">   40</a></span>&#160;<span class="keyword">class </span><a class="code" href="classoperations__research_1_1_christofides_path_solver.html">ChristofidesPathSolver</a> {</div>
<div class="line"><a name="l00041"></a><span class="lineno">   41</span>&#160; <span class="keyword">public</span>:</div>
<div class="line"><a name="l00042"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_christofides_path_solver.html#a1d4f082de5fc3eed348d65eb30b5f3e7">   42</a></span>&#160;  <span class="keyword">enum class</span> <a class="code" href="classoperations__research_1_1_christofides_path_solver.html#a1d4f082de5fc3eed348d65eb30b5f3e7">MatchingAlgorithm</a> {</div>
<div class="line"><a name="l00043"></a><span class="lineno">   43</span>&#160;    <a class="code" href="classoperations__research_1_1_christofides_path_solver.html#a1d4f082de5fc3eed348d65eb30b5f3e7ab66d0823917c9351a4cb68dff77f445a">MINIMUM_WEIGHT_MATCHING</a>,</div>
<div class="line"><a name="l00044"></a><span class="lineno">   44</span>&#160;<span class="preprocessor">#if defined(USE_CBC) || defined(USE_SCIP)</span></div>
<div class="line"><a name="l00045"></a><span class="lineno">   45</span>&#160;    <a class="code" href="classoperations__research_1_1_christofides_path_solver.html#a1d4f082de5fc3eed348d65eb30b5f3e7a201b88f6589fa1271207fe29f583dc96">MINIMUM_WEIGHT_MATCHING_WITH_MIP</a>,</div>
<div class="line"><a name="l00046"></a><span class="lineno">   46</span>&#160;<span class="preprocessor">#endif  // defined(USE_CBC) || defined(USE_SCIP)</span></div>
<div class="line"><a name="l00047"></a><span class="lineno">   47</span>&#160;    <a class="code" href="classoperations__research_1_1_christofides_path_solver.html#a1d4f082de5fc3eed348d65eb30b5f3e7a99c5fe202c37dcd8ed9cc60926a4f525">MINIMAL_WEIGHT_MATCHING</a>,</div>
<div class="line"><a name="l00048"></a><span class="lineno">   48</span>&#160;  };</div>
<div class="line"><a name="l00049"></a><span class="lineno">   49</span>&#160;  <a class="code" href="classoperations__research_1_1_christofides_path_solver.html#a87145cf1f5a36d27fd856596a23d495a">ChristofidesPathSolver</a>(<a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> num_nodes, CostFunction costs);</div>
<div class="line"><a name="l00050"></a><span class="lineno">   50</span>&#160; </div>
<div class="line"><a name="l00051"></a><span class="lineno">   51</span>&#160;  <span class="comment">// Sets the matching algorith to use. A minimum weight perfect matching</span></div>
<div class="line"><a name="l00052"></a><span class="lineno">   52</span>&#160;  <span class="comment">// (MINIMUM_WEIGHT_MATCHING) guarantees the 3/2 upper bound to the optimal</span></div>
<div class="line"><a name="l00053"></a><span class="lineno">   53</span>&#160;  <span class="comment">// solution. A minimal weight perfect matching (MINIMAL_WEIGHT_MATCHING)</span></div>
<div class="line"><a name="l00054"></a><span class="lineno">   54</span>&#160;  <span class="comment">// finds a locally minimal weight matching which does not offer any bound</span></div>
<div class="line"><a name="l00055"></a><span class="lineno">   55</span>&#160;  <span class="comment">// guarantee but, as of 1/2017, is orders of magnitude faster than the</span></div>
<div class="line"><a name="l00056"></a><span class="lineno">   56</span>&#160;  <span class="comment">// minimum matching.</span></div>
<div class="line"><a name="l00057"></a><span class="lineno">   57</span>&#160;  <span class="comment">// By default, MINIMAL_WEIGHT_MATCHING is selected.</span></div>
<div class="line"><a name="l00058"></a><span class="lineno">   58</span>&#160;  <span class="comment">// TODO(user): Change the default when minimum matching gets faster.</span></div>
<div class="line"><a name="l00059"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_christofides_path_solver.html#ae2b64dddd58baf1bc5e01cb14971e52e">   59</a></span>&#160;  <span class="keywordtype">void</span> <a class="code" href="classoperations__research_1_1_christofides_path_solver.html#ae2b64dddd58baf1bc5e01cb14971e52e">SetMatchingAlgorithm</a>(<a class="code" href="classoperations__research_1_1_christofides_path_solver.html#a1d4f082de5fc3eed348d65eb30b5f3e7">MatchingAlgorithm</a> matching) {</div>
<div class="line"><a name="l00060"></a><span class="lineno">   60</span>&#160;    matching_ = matching;</div>
<div class="line"><a name="l00061"></a><span class="lineno">   61</span>&#160;  }</div>
<div class="line"><a name="l00062"></a><span class="lineno">   62</span>&#160; </div>
<div class="line"><a name="l00063"></a><span class="lineno">   63</span>&#160;  <span class="comment">// Returns the cost of the approximate TSP tour.</span></div>
<div class="line"><a name="l00064"></a><span class="lineno">   64</span>&#160;  CostType <a class="code" href="classoperations__research_1_1_christofides_path_solver.html#ab558649a26fef3a74f0909ef5af45e90">TravelingSalesmanCost</a>();</div>
<div class="line"><a name="l00065"></a><span class="lineno">   65</span>&#160; </div>
<div class="line"><a name="l00066"></a><span class="lineno">   66</span>&#160;  <span class="comment">// Returns the approximate TSP tour.</span></div>
<div class="line"><a name="l00067"></a><span class="lineno">   67</span>&#160;  std::vector&lt;NodeIndex&gt; <a class="code" href="classoperations__research_1_1_christofides_path_solver.html#a3076d9001536ea98d419faa81e7d8a47">TravelingSalesmanPath</a>();</div>
<div class="line"><a name="l00068"></a><span class="lineno">   68</span>&#160; </div>
<div class="line"><a name="l00069"></a><span class="lineno">   69</span>&#160; <span class="keyword">private</span>:</div>
<div class="line"><a name="l00070"></a><span class="lineno">   70</span>&#160;  <span class="comment">// Runs the Christofides algorithm.</span></div>
<div class="line"><a name="l00071"></a><span class="lineno">   71</span>&#160;  <span class="keywordtype">void</span> Solve();</div>
<div class="line"><a name="l00072"></a><span class="lineno">   72</span>&#160; </div>
<div class="line"><a name="l00073"></a><span class="lineno">   73</span>&#160;  <span class="comment">// Safe addition operator to avoid overflows when possible.</span></div>
<div class="line"><a name="l00074"></a><span class="lineno">   74</span>&#160;  <span class="comment">// template &lt;typename T&gt;</span></div>
<div class="line"><a name="l00075"></a><span class="lineno">   75</span>&#160;  <span class="comment">// T SafeAdd(T a, T b) {</span></div>
<div class="line"><a name="l00076"></a><span class="lineno">   76</span>&#160;  <span class="comment">//   return a + b;</span></div>
<div class="line"><a name="l00077"></a><span class="lineno">   77</span>&#160;  <span class="comment">// }</span></div>
<div class="line"><a name="l00078"></a><span class="lineno">   78</span>&#160;  <span class="comment">// template &lt;&gt;</span></div>
<div class="line"><a name="l00079"></a><span class="lineno">   79</span>&#160;  int64 SafeAdd(int64 a, int64 b) { <span class="keywordflow">return</span> CapAdd(a, b); }</div>
<div class="line"><a name="l00080"></a><span class="lineno">   80</span>&#160; </div>
<div class="line"><a name="l00081"></a><span class="lineno">   81</span>&#160;  <span class="comment">// Matching algorithm to use.</span></div>
<div class="line"><a name="l00082"></a><span class="lineno">   82</span>&#160;  <a class="code" href="classoperations__research_1_1_christofides_path_solver.html#a1d4f082de5fc3eed348d65eb30b5f3e7">MatchingAlgorithm</a> matching_;</div>
<div class="line"><a name="l00083"></a><span class="lineno">   83</span>&#160; </div>
<div class="line"><a name="l00084"></a><span class="lineno">   84</span>&#160;  <span class="comment">// The complete graph on the nodes of the problem.</span></div>
<div class="line"><a name="l00085"></a><span class="lineno">   85</span>&#160;  CompleteGraph&lt;NodeIndex, ArcIndex&gt; graph_;</div>
<div class="line"><a name="l00086"></a><span class="lineno">   86</span>&#160; </div>
<div class="line"><a name="l00087"></a><span class="lineno">   87</span>&#160;  <span class="comment">// Function returning the cost between nodes of the problem.</span></div>
<div class="line"><a name="l00088"></a><span class="lineno">   88</span>&#160;  <span class="keyword">const</span> CostFunction costs_;</div>
<div class="line"><a name="l00089"></a><span class="lineno">   89</span>&#160; </div>
<div class="line"><a name="l00090"></a><span class="lineno">   90</span>&#160;  <span class="comment">// The cost of the computed TSP path.</span></div>
<div class="line"><a name="l00091"></a><span class="lineno">   91</span>&#160;  CostType tsp_cost_;</div>
<div class="line"><a name="l00092"></a><span class="lineno">   92</span>&#160; </div>
<div class="line"><a name="l00093"></a><span class="lineno">   93</span>&#160;  <span class="comment">// The path of the computed TSP,</span></div>
<div class="line"><a name="l00094"></a><span class="lineno">   94</span>&#160;  std::vector&lt;NodeIndex&gt; tsp_path_;</div>
<div class="line"><a name="l00095"></a><span class="lineno">   95</span>&#160; </div>
<div class="line"><a name="l00096"></a><span class="lineno">   96</span>&#160;  <span class="comment">// True if the TSP has been solved, false otherwise.</span></div>
<div class="line"><a name="l00097"></a><span class="lineno">   97</span>&#160;  <span class="keywordtype">bool</span> solved_;</div>
<div class="line"><a name="l00098"></a><span class="lineno">   98</span>&#160;};</div>
<div class="line"><a name="l00099"></a><span class="lineno">   99</span>&#160; </div>
<div class="line"><a name="l00100"></a><span class="lineno">  100</span>&#160;<span class="comment">// Computes a minimum weight perfect matching on an undirected graph.</span></div>
<div class="line"><a name="l00101"></a><span class="lineno">  101</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> WeightFunctionType, <span class="keyword">typename</span> GraphType&gt;</div>
<div class="line"><a name="l00102"></a><span class="lineno">  102</span>&#160;std::vector&lt;</div>
<div class="line"><a name="l00103"></a><span class="lineno">  103</span>&#160;    std::pair&lt;typename GraphType::NodeIndex, typename GraphType::NodeIndex&gt;&gt;</div>
<div class="line"><a name="l00104"></a><span class="lineno"><a class="line" href="namespaceoperations__research.html#a3174292e878ea613e33ac86334fde459">  104</a></span>&#160;<a class="code" href="namespaceoperations__research.html#a3174292e878ea613e33ac86334fde459">ComputeMinimumWeightMatching</a>(<span class="keyword">const</span> GraphType&amp; graph,</div>
<div class="line"><a name="l00105"></a><span class="lineno">  105</span>&#160;                             <span class="keyword">const</span> WeightFunctionType&amp; weight) {</div>
<div class="line"><a name="l00106"></a><span class="lineno">  106</span>&#160;  <span class="keyword">using</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> = <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">GraphType::ArcIndex</a>;</div>
<div class="line"><a name="l00107"></a><span class="lineno">  107</span>&#160;  <span class="keyword">using</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> = <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">GraphType::NodeIndex</a>;</div>
<div class="line"><a name="l00108"></a><span class="lineno">  108</span>&#160;  MinCostPerfectMatching matching(graph.num_nodes());</div>
<div class="line"><a name="l00109"></a><span class="lineno">  109</span>&#160;  <span class="keywordflow">for</span> (<a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> tail : graph.AllNodes()) {</div>
<div class="line"><a name="l00110"></a><span class="lineno">  110</span>&#160;    <span class="keywordflow">for</span> (<span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> arc : graph.OutgoingArcs(tail)) {</div>
<div class="line"><a name="l00111"></a><span class="lineno">  111</span>&#160;      <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> head = graph.Head(arc);</div>
<div class="line"><a name="l00112"></a><span class="lineno">  112</span>&#160;      <span class="comment">// Adding both arcs is redudant for MinCostPerfectMatching.</span></div>
<div class="line"><a name="l00113"></a><span class="lineno">  113</span>&#160;      <span class="keywordflow">if</span> (tail &lt; head) {</div>
<div class="line"><a name="l00114"></a><span class="lineno">  114</span>&#160;        matching.AddEdgeWithCost(tail, head, weight(arc));</div>
<div class="line"><a name="l00115"></a><span class="lineno">  115</span>&#160;      }</div>
<div class="line"><a name="l00116"></a><span class="lineno">  116</span>&#160;    }</div>
<div class="line"><a name="l00117"></a><span class="lineno">  117</span>&#160;  }</div>
<div class="line"><a name="l00118"></a><span class="lineno">  118</span>&#160;  MinCostPerfectMatching::Status status = matching.Solve();</div>
<div class="line"><a name="l00119"></a><span class="lineno">  119</span>&#160;  DCHECK_EQ(status, MinCostPerfectMatching::OPTIMAL);</div>
<div class="line"><a name="l00120"></a><span class="lineno">  120</span>&#160;  std::vector&lt;std::pair&lt;NodeIndex, NodeIndex&gt;&gt; match;</div>
<div class="line"><a name="l00121"></a><span class="lineno">  121</span>&#160;  <span class="keywordflow">for</span> (<a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> tail : graph.AllNodes()) {</div>
<div class="line"><a name="l00122"></a><span class="lineno">  122</span>&#160;    <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> head = matching.Match(tail);</div>
<div class="line"><a name="l00123"></a><span class="lineno">  123</span>&#160;    <span class="keywordflow">if</span> (tail &lt; head) {  <span class="comment">// Both arcs are matched for a given edge, we keep one.</span></div>
<div class="line"><a name="l00124"></a><span class="lineno">  124</span>&#160;      match.emplace_back(tail, head);</div>
<div class="line"><a name="l00125"></a><span class="lineno">  125</span>&#160;    }</div>
<div class="line"><a name="l00126"></a><span class="lineno">  126</span>&#160;  }</div>
<div class="line"><a name="l00127"></a><span class="lineno">  127</span>&#160;  <span class="keywordflow">return</span> match;</div>
<div class="line"><a name="l00128"></a><span class="lineno">  128</span>&#160;}</div>
<div class="line"><a name="l00129"></a><span class="lineno">  129</span>&#160; </div>
<div class="line"><a name="l00130"></a><span class="lineno">  130</span>&#160;<span class="preprocessor">#if defined(USE_CBC) || defined(USE_SCIP)</span></div>
<div class="line"><a name="l00131"></a><span class="lineno">  131</span>&#160;<span class="comment">// Computes a minimum weight perfect matching on an undirected graph using a</span></div>
<div class="line"><a name="l00132"></a><span class="lineno">  132</span>&#160;<span class="comment">// Mixed Integer Programming model.</span></div>
<div class="line"><a name="l00133"></a><span class="lineno">  133</span>&#160;<span class="comment">// TODO(user): Handle infeasible cases if this algorithm is used outside of</span></div>
<div class="line"><a name="l00134"></a><span class="lineno">  134</span>&#160;<span class="comment">// Christofides.</span></div>
<div class="line"><a name="l00135"></a><span class="lineno">  135</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> WeightFunctionType, <span class="keyword">typename</span> GraphType&gt;</div>
<div class="line"><a name="l00136"></a><span class="lineno">  136</span>&#160;std::vector&lt;</div>
<div class="line"><a name="l00137"></a><span class="lineno">  137</span>&#160;    std::pair&lt;typename GraphType::NodeIndex, typename GraphType::NodeIndex&gt;&gt;</div>
<div class="line"><a name="l00138"></a><span class="lineno"><a class="line" href="namespaceoperations__research.html#a48f6892917d77639f011cd780ce88bde">  138</a></span>&#160;<a class="code" href="namespaceoperations__research.html#a48f6892917d77639f011cd780ce88bde">ComputeMinimumWeightMatchingWithMIP</a>(<span class="keyword">const</span> GraphType&amp; graph,</div>
<div class="line"><a name="l00139"></a><span class="lineno">  139</span>&#160;                                    <span class="keyword">const</span> WeightFunctionType&amp; weight) {</div>
<div class="line"><a name="l00140"></a><span class="lineno">  140</span>&#160;  <span class="keyword">using</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> = <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">GraphType::ArcIndex</a>;</div>
<div class="line"><a name="l00141"></a><span class="lineno">  141</span>&#160;  <span class="keyword">using</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> = <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">GraphType::NodeIndex</a>;</div>
<div class="line"><a name="l00142"></a><span class="lineno">  142</span>&#160;  MPModelProto model;</div>
<div class="line"><a name="l00143"></a><span class="lineno">  143</span>&#160;  model.set_maximize(<span class="keyword">false</span>);</div>
<div class="line"><a name="l00144"></a><span class="lineno">  144</span>&#160;  <span class="comment">// The model is composed of Boolean decision variables to select matching arcs</span></div>
<div class="line"><a name="l00145"></a><span class="lineno">  145</span>&#160;  <span class="comment">// and constraints ensuring that each node appears in exactly one selected</span></div>
<div class="line"><a name="l00146"></a><span class="lineno">  146</span>&#160;  <span class="comment">// arc. The objective is to minimize the sum of the weights of selected arcs.</span></div>
<div class="line"><a name="l00147"></a><span class="lineno">  147</span>&#160;  <span class="comment">// It is assumed the graph is symmetrical.</span></div>
<div class="line"><a name="l00148"></a><span class="lineno">  148</span>&#160;  std::vector&lt;int&gt; variable_indices(graph.num_arcs(), -1);</div>
<div class="line"><a name="l00149"></a><span class="lineno">  149</span>&#160;  <span class="keywordflow">for</span> (<a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> node : graph.AllNodes()) {</div>
<div class="line"><a name="l00150"></a><span class="lineno">  150</span>&#160;    <span class="comment">// Creating arc-selection Boolean variable.</span></div>
<div class="line"><a name="l00151"></a><span class="lineno">  151</span>&#160;    <span class="keywordflow">for</span> (<span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> arc : graph.OutgoingArcs(node)) {</div>
<div class="line"><a name="l00152"></a><span class="lineno">  152</span>&#160;      <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> head = graph.Head(arc);</div>
<div class="line"><a name="l00153"></a><span class="lineno">  153</span>&#160;      <span class="keywordflow">if</span> (node &lt; head) {</div>
<div class="line"><a name="l00154"></a><span class="lineno">  154</span>&#160;        variable_indices[arc] = model.variable_size();</div>
<div class="line"><a name="l00155"></a><span class="lineno">  155</span>&#160;        MPVariableProto* <span class="keyword">const</span> arc_var = model.add_variable();</div>
<div class="line"><a name="l00156"></a><span class="lineno">  156</span>&#160;        arc_var-&gt;set_lower_bound(0);</div>
<div class="line"><a name="l00157"></a><span class="lineno">  157</span>&#160;        arc_var-&gt;set_upper_bound(1);</div>
<div class="line"><a name="l00158"></a><span class="lineno">  158</span>&#160;        arc_var-&gt;set_is_integer(<span class="keyword">true</span>);</div>
<div class="line"><a name="l00159"></a><span class="lineno">  159</span>&#160;        arc_var-&gt;set_objective_coefficient(weight(arc));</div>
<div class="line"><a name="l00160"></a><span class="lineno">  160</span>&#160;      }</div>
<div class="line"><a name="l00161"></a><span class="lineno">  161</span>&#160;    }</div>
<div class="line"><a name="l00162"></a><span class="lineno">  162</span>&#160;    <span class="comment">// Creating matching constraint:</span></div>
<div class="line"><a name="l00163"></a><span class="lineno">  163</span>&#160;    <span class="comment">// for all node i, sum(j) arc(i,j) + sum(j) arc(j,i) = 1</span></div>
<div class="line"><a name="l00164"></a><span class="lineno">  164</span>&#160;    MPConstraintProto* <span class="keyword">const</span> one_of_ct = model.add_constraint();</div>
<div class="line"><a name="l00165"></a><span class="lineno">  165</span>&#160;    one_of_ct-&gt;set_lower_bound(1);</div>
<div class="line"><a name="l00166"></a><span class="lineno">  166</span>&#160;    one_of_ct-&gt;set_upper_bound(1);</div>
<div class="line"><a name="l00167"></a><span class="lineno">  167</span>&#160;  }</div>
<div class="line"><a name="l00168"></a><span class="lineno">  168</span>&#160;  <span class="keywordflow">for</span> (<a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> node : graph.AllNodes()) {</div>
<div class="line"><a name="l00169"></a><span class="lineno">  169</span>&#160;    <span class="keywordflow">for</span> (<span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> arc : graph.OutgoingArcs(node)) {</div>
<div class="line"><a name="l00170"></a><span class="lineno">  170</span>&#160;      <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> head = graph.Head(arc);</div>
<div class="line"><a name="l00171"></a><span class="lineno">  171</span>&#160;      <span class="keywordflow">if</span> (node &lt; head) {</div>
<div class="line"><a name="l00172"></a><span class="lineno">  172</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">int</span> arc_var = variable_indices[arc];</div>
<div class="line"><a name="l00173"></a><span class="lineno">  173</span>&#160;        DCHECK_GE(arc_var, 0);</div>
<div class="line"><a name="l00174"></a><span class="lineno">  174</span>&#160;        MPConstraintProto* one_of_ct = model.mutable_constraint(node);</div>
<div class="line"><a name="l00175"></a><span class="lineno">  175</span>&#160;        one_of_ct-&gt;add_var_index(arc_var);</div>
<div class="line"><a name="l00176"></a><span class="lineno">  176</span>&#160;        one_of_ct-&gt;add_coefficient(1);</div>
<div class="line"><a name="l00177"></a><span class="lineno">  177</span>&#160;        one_of_ct = model.mutable_constraint(head);</div>
<div class="line"><a name="l00178"></a><span class="lineno">  178</span>&#160;        one_of_ct-&gt;add_var_index(arc_var);</div>
<div class="line"><a name="l00179"></a><span class="lineno">  179</span>&#160;        one_of_ct-&gt;add_coefficient(1);</div>
<div class="line"><a name="l00180"></a><span class="lineno">  180</span>&#160;      }</div>
<div class="line"><a name="l00181"></a><span class="lineno">  181</span>&#160;    }</div>
<div class="line"><a name="l00182"></a><span class="lineno">  182</span>&#160;  }</div>
<div class="line"><a name="l00183"></a><span class="lineno">  183</span>&#160;<span class="preprocessor">#if defined(USE_SCIP)</span></div>
<div class="line"><a name="l00184"></a><span class="lineno">  184</span>&#160;  MPSolver mp_solver(<span class="stringliteral">&quot;MatchingWithSCIP&quot;</span>,</div>
<div class="line"><a name="l00185"></a><span class="lineno">  185</span>&#160;                     MPSolver::SCIP_MIXED_INTEGER_PROGRAMMING);</div>
<div class="line"><a name="l00186"></a><span class="lineno">  186</span>&#160;<span class="preprocessor">#elif defined(USE_CBC)</span></div>
<div class="line"><a name="l00187"></a><span class="lineno">  187</span>&#160;  MPSolver mp_solver(<span class="stringliteral">&quot;MatchingWithCBC&quot;</span>,</div>
<div class="line"><a name="l00188"></a><span class="lineno">  188</span>&#160;                     MPSolver::CBC_MIXED_INTEGER_PROGRAMMING);</div>
<div class="line"><a name="l00189"></a><span class="lineno">  189</span>&#160;<span class="preprocessor">#endif</span></div>
<div class="line"><a name="l00190"></a><span class="lineno">  190</span>&#160;  std::string error;</div>
<div class="line"><a name="l00191"></a><span class="lineno">  191</span>&#160;  mp_solver.LoadModelFromProto(model, &amp;error);</div>
<div class="line"><a name="l00192"></a><span class="lineno">  192</span>&#160;  MPSolver::ResultStatus status = mp_solver.Solve();</div>
<div class="line"><a name="l00193"></a><span class="lineno">  193</span>&#160;  CHECK_EQ(status, MPSolver::OPTIMAL);</div>
<div class="line"><a name="l00194"></a><span class="lineno">  194</span>&#160;  MPSolutionResponse response;</div>
<div class="line"><a name="l00195"></a><span class="lineno">  195</span>&#160;  mp_solver.FillSolutionResponseProto(&amp;response);</div>
<div class="line"><a name="l00196"></a><span class="lineno">  196</span>&#160;  std::vector&lt;std::pair&lt;NodeIndex, NodeIndex&gt;&gt; matching;</div>
<div class="line"><a name="l00197"></a><span class="lineno">  197</span>&#160;  <span class="keywordflow">for</span> (<a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> arc = 0; arc &lt; variable_indices.size(); ++arc) {</div>
<div class="line"><a name="l00198"></a><span class="lineno">  198</span>&#160;    <span class="keyword">const</span> <span class="keywordtype">int</span> arc_var = variable_indices[arc];</div>
<div class="line"><a name="l00199"></a><span class="lineno">  199</span>&#160;    <span class="keywordflow">if</span> (arc_var &gt;= 0 &amp;&amp; response.variable_value(arc_var) &gt; .9) {</div>
<div class="line"><a name="l00200"></a><span class="lineno">  200</span>&#160;      DCHECK_GE(response.variable_value(arc_var), 1.0 - 1e-4);</div>
<div class="line"><a name="l00201"></a><span class="lineno">  201</span>&#160;      matching.emplace_back(graph.Tail(arc), graph.Head(arc));</div>
<div class="line"><a name="l00202"></a><span class="lineno">  202</span>&#160;    }</div>
<div class="line"><a name="l00203"></a><span class="lineno">  203</span>&#160;  }</div>
<div class="line"><a name="l00204"></a><span class="lineno">  204</span>&#160;  <span class="keywordflow">return</span> matching;</div>
<div class="line"><a name="l00205"></a><span class="lineno">  205</span>&#160;}</div>
<div class="line"><a name="l00206"></a><span class="lineno">  206</span>&#160;<span class="preprocessor">#endif  // defined(USE_CBC) || defined(USE_SCIP)</span></div>
<div class="line"><a name="l00207"></a><span class="lineno">  207</span>&#160; </div>
<div class="line"><a name="l00208"></a><span class="lineno">  208</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> CostType, <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a>, <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a>,</div>
<div class="line"><a name="l00209"></a><span class="lineno">  209</span>&#160;          <span class="keyword">typename</span> CostFunction&gt;</div>
<div class="line"><a name="l00210"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_christofides_path_solver.html#a87145cf1f5a36d27fd856596a23d495a">  210</a></span>&#160;<a class="code" href="classoperations__research_1_1_christofides_path_solver.html#a87145cf1f5a36d27fd856596a23d495a">ChristofidesPathSolver&lt;CostType, ArcIndex, NodeIndex, CostFunction&gt;::</a></div>
<div class="line"><a name="l00211"></a><span class="lineno">  211</span>&#160;<a class="code" href="classoperations__research_1_1_christofides_path_solver.html#a87145cf1f5a36d27fd856596a23d495a">    ChristofidesPathSolver</a>(<a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> num_nodes, CostFunction costs)</div>
<div class="line"><a name="l00212"></a><span class="lineno">  212</span>&#160;    : matching_(<a class="code" href="classoperations__research_1_1_christofides_path_solver.html#a1d4f082de5fc3eed348d65eb30b5f3e7">MatchingAlgorithm</a>::MINIMAL_WEIGHT_MATCHING),</div>
<div class="line"><a name="l00213"></a><span class="lineno">  213</span>&#160;      graph_(num_nodes),</div>
<div class="line"><a name="l00214"></a><span class="lineno">  214</span>&#160;      costs_(std::move(costs)),</div>
<div class="line"><a name="l00215"></a><span class="lineno">  215</span>&#160;      tsp_cost_(0),</div>
<div class="line"><a name="l00216"></a><span class="lineno">  216</span>&#160;      solved_(false) {}</div>
<div class="line"><a name="l00217"></a><span class="lineno">  217</span>&#160; </div>
<div class="line"><a name="l00218"></a><span class="lineno">  218</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> CostType, <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a>, <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a>,</div>
<div class="line"><a name="l00219"></a><span class="lineno">  219</span>&#160;          <span class="keyword">typename</span> CostFunction&gt;</div>
<div class="line"><a name="l00220"></a><span class="lineno">  220</span>&#160;CostType <a class="code" href="classoperations__research_1_1_christofides_path_solver.html">ChristofidesPathSolver</a>&lt;CostType, <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a>, <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a>,</div>
<div class="line"><a name="l00221"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_christofides_path_solver.html#ab558649a26fef3a74f0909ef5af45e90">  221</a></span>&#160;                                CostFunction&gt;::TravelingSalesmanCost() {</div>
<div class="line"><a name="l00222"></a><span class="lineno">  222</span>&#160;  <span class="keywordflow">if</span> (!solved_) {</div>
<div class="line"><a name="l00223"></a><span class="lineno">  223</span>&#160;    Solve();</div>
<div class="line"><a name="l00224"></a><span class="lineno">  224</span>&#160;  }</div>
<div class="line"><a name="l00225"></a><span class="lineno">  225</span>&#160;  <span class="keywordflow">return</span> tsp_cost_;</div>
<div class="line"><a name="l00226"></a><span class="lineno">  226</span>&#160;}</div>
<div class="line"><a name="l00227"></a><span class="lineno">  227</span>&#160; </div>
<div class="line"><a name="l00228"></a><span class="lineno">  228</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> CostType, <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a>, <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a>,</div>
<div class="line"><a name="l00229"></a><span class="lineno">  229</span>&#160;          <span class="keyword">typename</span> CostFunction&gt;</div>
<div class="line"><a name="l00230"></a><span class="lineno">  230</span>&#160;std::vector&lt;NodeIndex&gt; <a class="code" href="classoperations__research_1_1_christofides_path_solver.html">ChristofidesPathSolver</a>&lt;</div>
<div class="line"><a name="l00231"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_christofides_path_solver.html#a3076d9001536ea98d419faa81e7d8a47">  231</a></span>&#160;    CostType, <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a>, <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a>, CostFunction&gt;::TravelingSalesmanPath() {</div>
<div class="line"><a name="l00232"></a><span class="lineno">  232</span>&#160;  <span class="keywordflow">if</span> (!solved_) {</div>
<div class="line"><a name="l00233"></a><span class="lineno">  233</span>&#160;    Solve();</div>
<div class="line"><a name="l00234"></a><span class="lineno">  234</span>&#160;  }</div>
<div class="line"><a name="l00235"></a><span class="lineno">  235</span>&#160;  <span class="keywordflow">return</span> tsp_path_;</div>
<div class="line"><a name="l00236"></a><span class="lineno">  236</span>&#160;}</div>
<div class="line"><a name="l00237"></a><span class="lineno">  237</span>&#160; </div>
<div class="line"><a name="l00238"></a><span class="lineno">  238</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> CostType, <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a>, <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a>,</div>
<div class="line"><a name="l00239"></a><span class="lineno">  239</span>&#160;          <span class="keyword">typename</span> CostFunction&gt;</div>
<div class="line"><a name="l00240"></a><span class="lineno">  240</span>&#160;<span class="keywordtype">void</span> <a class="code" href="classoperations__research_1_1_christofides_path_solver.html">ChristofidesPathSolver</a>&lt;CostType, <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a>, <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a>,</div>
<div class="line"><a name="l00241"></a><span class="lineno">  241</span>&#160;                            CostFunction&gt;::Solve() {</div>
<div class="line"><a name="l00242"></a><span class="lineno">  242</span>&#160;  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> num_nodes = graph_.num_nodes();</div>
<div class="line"><a name="l00243"></a><span class="lineno">  243</span>&#160;  tsp_path_.clear();</div>
<div class="line"><a name="l00244"></a><span class="lineno">  244</span>&#160;  tsp_cost_ = 0;</div>
<div class="line"><a name="l00245"></a><span class="lineno">  245</span>&#160;  <span class="keywordflow">if</span> (num_nodes == 1) {</div>
<div class="line"><a name="l00246"></a><span class="lineno">  246</span>&#160;    tsp_path_ = {0, 0};</div>
<div class="line"><a name="l00247"></a><span class="lineno">  247</span>&#160;  }</div>
<div class="line"><a name="l00248"></a><span class="lineno">  248</span>&#160;  <span class="keywordflow">if</span> (num_nodes &lt;= 1) {</div>
<div class="line"><a name="l00249"></a><span class="lineno">  249</span>&#160;    <span class="keywordflow">return</span>;</div>
<div class="line"><a name="l00250"></a><span class="lineno">  250</span>&#160;  }</div>
<div class="line"><a name="l00251"></a><span class="lineno">  251</span>&#160;  <span class="comment">// Compute Minimum Spanning Tree.</span></div>
<div class="line"><a name="l00252"></a><span class="lineno">  252</span>&#160;  <span class="keyword">const</span> std::vector&lt;ArcIndex&gt; mst =</div>
<div class="line"><a name="l00253"></a><span class="lineno">  253</span>&#160;      <a class="code" href="namespaceoperations__research.html#a33a2f4c26fd60cd0fa98257b571c974f">BuildPrimMinimumSpanningTree</a>(graph_, [<span class="keyword">this</span>](<a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> arc) {</div>
<div class="line"><a name="l00254"></a><span class="lineno">  254</span>&#160;        <span class="keywordflow">return</span> costs_(graph_.Tail(arc), graph_.Head(arc));</div>
<div class="line"><a name="l00255"></a><span class="lineno">  255</span>&#160;      });</div>
<div class="line"><a name="l00256"></a><span class="lineno">  256</span>&#160;  <span class="comment">// Detect odd degree nodes.</span></div>
<div class="line"><a name="l00257"></a><span class="lineno">  257</span>&#160;  std::vector&lt;NodeIndex&gt; degrees(num_nodes, 0);</div>
<div class="line"><a name="l00258"></a><span class="lineno">  258</span>&#160;  <span class="keywordflow">for</span> (<a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> arc : mst) {</div>
<div class="line"><a name="l00259"></a><span class="lineno">  259</span>&#160;    degrees[graph_.Tail(arc)]++;</div>
<div class="line"><a name="l00260"></a><span class="lineno">  260</span>&#160;    degrees[graph_.Head(arc)]++;</div>
<div class="line"><a name="l00261"></a><span class="lineno">  261</span>&#160;  }</div>
<div class="line"><a name="l00262"></a><span class="lineno">  262</span>&#160;  std::vector&lt;NodeIndex&gt; odd_degree_nodes;</div>
<div class="line"><a name="l00263"></a><span class="lineno">  263</span>&#160;  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; degrees.size(); ++i) {</div>
<div class="line"><a name="l00264"></a><span class="lineno">  264</span>&#160;    <span class="keywordflow">if</span> (degrees[i] % 2 != 0) {</div>
<div class="line"><a name="l00265"></a><span class="lineno">  265</span>&#160;      odd_degree_nodes.push_back(i);</div>
<div class="line"><a name="l00266"></a><span class="lineno">  266</span>&#160;    }</div>
<div class="line"><a name="l00267"></a><span class="lineno">  267</span>&#160;  }</div>
<div class="line"><a name="l00268"></a><span class="lineno">  268</span>&#160;  <span class="comment">// Find minimum-weight perfect matching on odd-degree-node complete graph.</span></div>
<div class="line"><a name="l00269"></a><span class="lineno">  269</span>&#160;  <span class="comment">// TODO(user): Make this code available as an independent algorithm.</span></div>
<div class="line"><a name="l00270"></a><span class="lineno">  270</span>&#160;  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> reduced_size = odd_degree_nodes.size();</div>
<div class="line"><a name="l00271"></a><span class="lineno">  271</span>&#160;  DCHECK_NE(0, reduced_size);</div>
<div class="line"><a name="l00272"></a><span class="lineno">  272</span>&#160;  CompleteGraph&lt;NodeIndex, ArcIndex&gt; reduced_graph(reduced_size);</div>
<div class="line"><a name="l00273"></a><span class="lineno">  273</span>&#160;  std::vector&lt;std::pair&lt;NodeIndex, NodeIndex&gt;&gt; closure_arcs;</div>
<div class="line"><a name="l00274"></a><span class="lineno">  274</span>&#160;  <span class="keywordflow">switch</span> (matching_) {</div>
<div class="line"><a name="l00275"></a><span class="lineno">  275</span>&#160;    <span class="keywordflow">case</span> MatchingAlgorithm::MINIMUM_WEIGHT_MATCHING: {</div>
<div class="line"><a name="l00276"></a><span class="lineno">  276</span>&#160;      closure_arcs = <a class="code" href="namespaceoperations__research.html#a3174292e878ea613e33ac86334fde459">ComputeMinimumWeightMatching</a>(</div>
<div class="line"><a name="l00277"></a><span class="lineno">  277</span>&#160;          reduced_graph, [<span class="keyword">this</span>, &amp;reduced_graph,</div>
<div class="line"><a name="l00278"></a><span class="lineno">  278</span>&#160;                          &amp;odd_degree_nodes](<a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">CompleteGraph&lt;&gt;::ArcIndex</a> arc) {</div>
<div class="line"><a name="l00279"></a><span class="lineno">  279</span>&#160;            <span class="keywordflow">return</span> costs_(odd_degree_nodes[reduced_graph.Tail(arc)],</div>
<div class="line"><a name="l00280"></a><span class="lineno">  280</span>&#160;                          odd_degree_nodes[reduced_graph.Head(arc)]);</div>
<div class="line"><a name="l00281"></a><span class="lineno">  281</span>&#160;          });</div>
<div class="line"><a name="l00282"></a><span class="lineno">  282</span>&#160;      <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00283"></a><span class="lineno">  283</span>&#160;    }</div>
<div class="line"><a name="l00284"></a><span class="lineno">  284</span>&#160;<span class="preprocessor">#if defined(USE_CBC) || defined(USE_SCIP)</span></div>
<div class="line"><a name="l00285"></a><span class="lineno">  285</span>&#160;    <span class="keywordflow">case</span> MatchingAlgorithm::MINIMUM_WEIGHT_MATCHING_WITH_MIP: {</div>
<div class="line"><a name="l00286"></a><span class="lineno">  286</span>&#160;      closure_arcs = <a class="code" href="namespaceoperations__research.html#a48f6892917d77639f011cd780ce88bde">ComputeMinimumWeightMatchingWithMIP</a>(</div>
<div class="line"><a name="l00287"></a><span class="lineno">  287</span>&#160;          reduced_graph, [<span class="keyword">this</span>, &amp;reduced_graph,</div>
<div class="line"><a name="l00288"></a><span class="lineno">  288</span>&#160;                          &amp;odd_degree_nodes](<a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">CompleteGraph&lt;&gt;::ArcIndex</a> arc) {</div>
<div class="line"><a name="l00289"></a><span class="lineno">  289</span>&#160;            <span class="keywordflow">return</span> costs_(odd_degree_nodes[reduced_graph.Tail(arc)],</div>
<div class="line"><a name="l00290"></a><span class="lineno">  290</span>&#160;                          odd_degree_nodes[reduced_graph.Head(arc)]);</div>
<div class="line"><a name="l00291"></a><span class="lineno">  291</span>&#160;          });</div>
<div class="line"><a name="l00292"></a><span class="lineno">  292</span>&#160;      <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00293"></a><span class="lineno">  293</span>&#160;    }</div>
<div class="line"><a name="l00294"></a><span class="lineno">  294</span>&#160;<span class="preprocessor">#endif  // defined(USE_CBC) || defined(USE_SCIP)</span></div>
<div class="line"><a name="l00295"></a><span class="lineno">  295</span>&#160;    <span class="keywordflow">case</span> MatchingAlgorithm::MINIMAL_WEIGHT_MATCHING: {</div>
<div class="line"><a name="l00296"></a><span class="lineno">  296</span>&#160;      <span class="comment">// TODO(user): Cost caching was added and can gain up to 20% but</span></div>
<div class="line"><a name="l00297"></a><span class="lineno">  297</span>&#160;      <span class="comment">// increases memory usage; see if we can avoid caching.</span></div>
<div class="line"><a name="l00298"></a><span class="lineno">  298</span>&#160;      std::vector&lt;ArcIndex&gt; ordered_arcs(reduced_graph.num_arcs());</div>
<div class="line"><a name="l00299"></a><span class="lineno">  299</span>&#160;      std::vector&lt;CostType&gt; ordered_arc_costs(reduced_graph.num_arcs(), 0);</div>
<div class="line"><a name="l00300"></a><span class="lineno">  300</span>&#160;      <span class="keywordflow">for</span> (<span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> arc : reduced_graph.AllForwardArcs()) {</div>
<div class="line"><a name="l00301"></a><span class="lineno">  301</span>&#160;        ordered_arcs[arc] = arc;</div>
<div class="line"><a name="l00302"></a><span class="lineno">  302</span>&#160;        ordered_arc_costs[arc] =</div>
<div class="line"><a name="l00303"></a><span class="lineno">  303</span>&#160;            costs_(odd_degree_nodes[reduced_graph.Tail(arc)],</div>
<div class="line"><a name="l00304"></a><span class="lineno">  304</span>&#160;                   odd_degree_nodes[reduced_graph.Head(arc)]);</div>
<div class="line"><a name="l00305"></a><span class="lineno">  305</span>&#160;      }</div>
<div class="line"><a name="l00306"></a><span class="lineno">  306</span>&#160;      std::sort(ordered_arcs.begin(), ordered_arcs.end(),</div>
<div class="line"><a name="l00307"></a><span class="lineno">  307</span>&#160;                [&amp;ordered_arc_costs](<a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> arc_a, <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> arc_b) {</div>
<div class="line"><a name="l00308"></a><span class="lineno">  308</span>&#160;                  return ordered_arc_costs[arc_a] &lt; ordered_arc_costs[arc_b];</div>
<div class="line"><a name="l00309"></a><span class="lineno">  309</span>&#160;                });</div>
<div class="line"><a name="l00310"></a><span class="lineno">  310</span>&#160;      std::vector&lt;bool&gt; touched_nodes(reduced_size, <span class="keyword">false</span>);</div>
<div class="line"><a name="l00311"></a><span class="lineno">  311</span>&#160;      <span class="keywordflow">for</span> (<a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> arc_index = 0; closure_arcs.size() * 2 &lt; reduced_size;</div>
<div class="line"><a name="l00312"></a><span class="lineno">  312</span>&#160;           ++arc_index) {</div>
<div class="line"><a name="l00313"></a><span class="lineno">  313</span>&#160;        <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> arc = ordered_arcs[arc_index];</div>
<div class="line"><a name="l00314"></a><span class="lineno">  314</span>&#160;        <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> tail = reduced_graph.Tail(arc);</div>
<div class="line"><a name="l00315"></a><span class="lineno">  315</span>&#160;        <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> head = reduced_graph.Head(arc);</div>
<div class="line"><a name="l00316"></a><span class="lineno">  316</span>&#160;        <span class="keywordflow">if</span> (head != tail &amp;&amp; !touched_nodes[tail] &amp;&amp; !touched_nodes[head]) {</div>
<div class="line"><a name="l00317"></a><span class="lineno">  317</span>&#160;          touched_nodes[tail] = <span class="keyword">true</span>;</div>
<div class="line"><a name="l00318"></a><span class="lineno">  318</span>&#160;          touched_nodes[head] = <span class="keyword">true</span>;</div>
<div class="line"><a name="l00319"></a><span class="lineno">  319</span>&#160;          closure_arcs.emplace_back(tail, head);</div>
<div class="line"><a name="l00320"></a><span class="lineno">  320</span>&#160;        }</div>
<div class="line"><a name="l00321"></a><span class="lineno">  321</span>&#160;      }</div>
<div class="line"><a name="l00322"></a><span class="lineno">  322</span>&#160;      <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00323"></a><span class="lineno">  323</span>&#160;    }</div>
<div class="line"><a name="l00324"></a><span class="lineno">  324</span>&#160;  }</div>
<div class="line"><a name="l00325"></a><span class="lineno">  325</span>&#160;  <span class="comment">// Build Eulerian path on minimum spanning tree + closing edges from matching</span></div>
<div class="line"><a name="l00326"></a><span class="lineno">  326</span>&#160;  <span class="comment">// and extract a solution to the Traveling Salesman from the path by skipping</span></div>
<div class="line"><a name="l00327"></a><span class="lineno">  327</span>&#160;  <span class="comment">// duplicate nodes.</span></div>
<div class="line"><a name="l00328"></a><span class="lineno">  328</span>&#160;  <a class="code" href="classutil_1_1_reverse_arc_list_graph.html">::util::ReverseArcListGraph&lt;NodeIndex, ArcIndex&gt;</a> egraph(</div>
<div class="line"><a name="l00329"></a><span class="lineno">  329</span>&#160;      num_nodes, closure_arcs.size() + mst.size());</div>
<div class="line"><a name="l00330"></a><span class="lineno">  330</span>&#160;  <span class="keywordflow">for</span> (<a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> arc : mst) {</div>
<div class="line"><a name="l00331"></a><span class="lineno">  331</span>&#160;    egraph.AddArc(graph_.Tail(arc), graph_.Head(arc));</div>
<div class="line"><a name="l00332"></a><span class="lineno">  332</span>&#160;  }</div>
<div class="line"><a name="l00333"></a><span class="lineno">  333</span>&#160;  <span class="keywordflow">for</span> (<span class="keyword">const</span> <span class="keyword">auto</span> arc : closure_arcs) {</div>
<div class="line"><a name="l00334"></a><span class="lineno">  334</span>&#160;    egraph.AddArc(odd_degree_nodes[arc.first], odd_degree_nodes[arc.second]);</div>
<div class="line"><a name="l00335"></a><span class="lineno">  335</span>&#160;  }</div>
<div class="line"><a name="l00336"></a><span class="lineno">  336</span>&#160;  std::vector&lt;bool&gt; touched(num_nodes, <span class="keyword">false</span>);</div>
<div class="line"><a name="l00337"></a><span class="lineno">  337</span>&#160;  DCHECK(<a class="code" href="namespaceoperations__research.html#ab1cf773de0cae72d0c44efe5b8f4bb89">IsEulerianGraph</a>(egraph));</div>
<div class="line"><a name="l00338"></a><span class="lineno">  338</span>&#160;  <span class="keywordflow">for</span> (<span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> node : <a class="code" href="namespaceoperations__research.html#aa63055860fc53f8eed56d23d2571c180">BuildEulerianTourFromNode</a>(egraph, 0)) {</div>
<div class="line"><a name="l00339"></a><span class="lineno">  339</span>&#160;    <span class="keywordflow">if</span> (touched[node]) <span class="keywordflow">continue</span>;</div>
<div class="line"><a name="l00340"></a><span class="lineno">  340</span>&#160;    touched[node] = <span class="keyword">true</span>;</div>
<div class="line"><a name="l00341"></a><span class="lineno">  341</span>&#160;    tsp_cost_ = SafeAdd(tsp_cost_,</div>
<div class="line"><a name="l00342"></a><span class="lineno">  342</span>&#160;                        tsp_path_.empty() ? 0 : costs_(tsp_path_.back(), node));</div>
<div class="line"><a name="l00343"></a><span class="lineno">  343</span>&#160;    tsp_path_.push_back(node);</div>
<div class="line"><a name="l00344"></a><span class="lineno">  344</span>&#160;  }</div>
<div class="line"><a name="l00345"></a><span class="lineno">  345</span>&#160;  tsp_cost_ =</div>
<div class="line"><a name="l00346"></a><span class="lineno">  346</span>&#160;      SafeAdd(tsp_cost_, tsp_path_.empty() ? 0 : costs_(tsp_path_.back(), 0));</div>
<div class="line"><a name="l00347"></a><span class="lineno">  347</span>&#160;  tsp_path_.push_back(0);</div>
<div class="line"><a name="l00348"></a><span class="lineno">  348</span>&#160;  solved_ = <span class="keyword">true</span>;</div>
<div class="line"><a name="l00349"></a><span class="lineno">  349</span>&#160;}</div>
<div class="line"><a name="l00350"></a><span class="lineno">  350</span>&#160;}  <span class="comment">// namespace operations_research</span></div>
<div class="line"><a name="l00351"></a><span class="lineno">  351</span>&#160; </div>
<div class="line"><a name="l00352"></a><span class="lineno">  352</span>&#160;<span class="preprocessor">#endif  // OR_TOOLS_GRAPH_CHRISTOFIDES_H_</span></div>
</div><!-- fragment --></div><!-- contents -->
</div><!-- doc-content -->
<div class="ttc" id="aeulerian__path_8h_html"><div class="ttname"><a href="eulerian__path_8h.html">eulerian_path.h</a></div></div>
<div class="ttc" id="aclassoperations__research_1_1_christofides_path_solver_html"><div class="ttname"><a href="classoperations__research_1_1_christofides_path_solver.html">operations_research::ChristofidesPathSolver</a></div><div class="ttdef"><b>Definition:</b> <a href="christofides_8h_source.html#l00040">christofides.h:40</a></div></div>
<div class="ttc" id="aclassoperations__research_1_1_christofides_path_solver_html_a87145cf1f5a36d27fd856596a23d495a"><div class="ttname"><a href="classoperations__research_1_1_christofides_path_solver.html#a87145cf1f5a36d27fd856596a23d495a">operations_research::ChristofidesPathSolver::ChristofidesPathSolver</a></div><div class="ttdeci">ChristofidesPathSolver(NodeIndex num_nodes, CostFunction costs)</div><div class="ttdef"><b>Definition:</b> <a href="christofides_8h_source.html#l00210">christofides.h:211</a></div></div>
<div class="ttc" id="anamespaceoperations__research_html_ab1cf773de0cae72d0c44efe5b8f4bb89"><div class="ttname"><a href="namespaceoperations__research.html#ab1cf773de0cae72d0c44efe5b8f4bb89">operations_research::IsEulerianGraph</a></div><div class="ttdeci">bool IsEulerianGraph(const Graph &amp;graph)</div><div class="ttdef"><b>Definition:</b> <a href="eulerian__path_8h_source.html#l00040">eulerian_path.h:40</a></div></div>
<div class="ttc" id="anamespaceoperations__research_html_a48f6892917d77639f011cd780ce88bde"><div class="ttname"><a href="namespaceoperations__research.html#a48f6892917d77639f011cd780ce88bde">operations_research::ComputeMinimumWeightMatchingWithMIP</a></div><div class="ttdeci">std::vector&lt; std::pair&lt; typename GraphType::NodeIndex, typename GraphType::NodeIndex &gt; &gt; ComputeMinimumWeightMatchingWithMIP(const GraphType &amp;graph, const WeightFunctionType &amp;weight)</div><div class="ttdef"><b>Definition:</b> <a href="christofides_8h_source.html#l00138">christofides.h:138</a></div></div>
<div class="ttc" id="aclassoperations__research_1_1_christofides_path_solver_html_a1d4f082de5fc3eed348d65eb30b5f3e7a99c5fe202c37dcd8ed9cc60926a4f525"><div class="ttname"><a href="classoperations__research_1_1_christofides_path_solver.html#a1d4f082de5fc3eed348d65eb30b5f3e7a99c5fe202c37dcd8ed9cc60926a4f525">operations_research::ChristofidesPathSolver::MatchingAlgorithm::MINIMAL_WEIGHT_MATCHING</a></div><div class="ttdeci">@ MINIMAL_WEIGHT_MATCHING</div></div>
<div class="ttc" id="anamespaceoperations__research_html"><div class="ttname"><a href="namespaceoperations__research.html">operations_research</a></div><div class="ttdef"><b>Definition:</b> <a href="christofides_8h_source.html#l00033">christofides.h:33</a></div></div>
<div class="ttc" id="aminimum__spanning__tree_8h_html"><div class="ttname"><a href="minimum__spanning__tree_8h.html">minimum_spanning_tree.h</a></div></div>
<div class="ttc" id="anamespaceoperations__research_html_a3174292e878ea613e33ac86334fde459"><div class="ttname"><a href="namespaceoperations__research.html#a3174292e878ea613e33ac86334fde459">operations_research::ComputeMinimumWeightMatching</a></div><div class="ttdeci">std::vector&lt; std::pair&lt; typename GraphType::NodeIndex, typename GraphType::NodeIndex &gt; &gt; ComputeMinimumWeightMatching(const GraphType &amp;graph, const WeightFunctionType &amp;weight)</div><div class="ttdef"><b>Definition:</b> <a href="christofides_8h_source.html#l00104">christofides.h:104</a></div></div>
<div class="ttc" id="anamespaceoperations__research_html_a33a2f4c26fd60cd0fa98257b571c974f"><div class="ttname"><a href="namespaceoperations__research.html#a33a2f4c26fd60cd0fa98257b571c974f">operations_research::BuildPrimMinimumSpanningTree</a></div><div class="ttdeci">std::vector&lt; typename Graph::ArcIndex &gt; BuildPrimMinimumSpanningTree(const Graph &amp;graph, const ArcValue &amp;arc_value)</div><div class="ttdef"><b>Definition:</b> <a href="minimum__spanning__tree_8h_source.html#l00115">minimum_spanning_tree.h:115</a></div></div>
<div class="ttc" id="aclassoperations__research_1_1_christofides_path_solver_html_ab558649a26fef3a74f0909ef5af45e90"><div class="ttname"><a href="classoperations__research_1_1_christofides_path_solver.html#ab558649a26fef3a74f0909ef5af45e90">operations_research::ChristofidesPathSolver::TravelingSalesmanCost</a></div><div class="ttdeci">CostType TravelingSalesmanCost()</div><div class="ttdef"><b>Definition:</b> <a href="christofides_8h_source.html#l00221">christofides.h:221</a></div></div>
<div class="ttc" id="aclassutil_1_1_reverse_arc_list_graph_html"><div class="ttname"><a href="classutil_1_1_reverse_arc_list_graph.html">util::ReverseArcListGraph</a></div><div class="ttdef"><b>Definition:</b> <a href="graph_8h_source.html#l00459">graph.h:460</a></div></div>
<div class="ttc" id="anamespaceoperations__research_html_ad90a9fc7d16bb2c98a088a4162bbbc9d"><div class="ttname"><a href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">operations_research::ArcIndex</a></div><div class="ttdeci">int32 ArcIndex</div><div class="ttdef"><b>Definition:</b> <a href="ebert__graph_8h_source.html#l00201">ebert_graph.h:201</a></div></div>
<div class="ttc" id="anamespaceoperations__research_html_aa63055860fc53f8eed56d23d2571c180"><div class="ttname"><a href="namespaceoperations__research.html#aa63055860fc53f8eed56d23d2571c180">operations_research::BuildEulerianTourFromNode</a></div><div class="ttdeci">std::vector&lt; NodeIndex &gt; BuildEulerianTourFromNode(const Graph &amp;graph, NodeIndex root)</div><div class="ttdef"><b>Definition:</b> <a href="eulerian__path_8h_source.html#l00116">eulerian_path.h:116</a></div></div>
<div class="ttc" id="aclassoperations__research_1_1_christofides_path_solver_html_a1d4f082de5fc3eed348d65eb30b5f3e7ab66d0823917c9351a4cb68dff77f445a"><div class="ttname"><a href="classoperations__research_1_1_christofides_path_solver.html#a1d4f082de5fc3eed348d65eb30b5f3e7ab66d0823917c9351a4cb68dff77f445a">operations_research::ChristofidesPathSolver::MatchingAlgorithm::MINIMUM_WEIGHT_MATCHING</a></div><div class="ttdeci">@ MINIMUM_WEIGHT_MATCHING</div></div>
<div class="ttc" id="agraph_8h_html"><div class="ttname"><a href="graph_8h.html">graph.h</a></div></div>
<div class="ttc" id="aclassoperations__research_1_1_christofides_path_solver_html_ae2b64dddd58baf1bc5e01cb14971e52e"><div class="ttname"><a href="classoperations__research_1_1_christofides_path_solver.html#ae2b64dddd58baf1bc5e01cb14971e52e">operations_research::ChristofidesPathSolver::SetMatchingAlgorithm</a></div><div class="ttdeci">void SetMatchingAlgorithm(MatchingAlgorithm matching)</div><div class="ttdef"><b>Definition:</b> <a href="christofides_8h_source.html#l00059">christofides.h:59</a></div></div>
<div class="ttc" id="aclassoperations__research_1_1_christofides_path_solver_html_a1d4f082de5fc3eed348d65eb30b5f3e7"><div class="ttname"><a href="classoperations__research_1_1_christofides_path_solver.html#a1d4f082de5fc3eed348d65eb30b5f3e7">operations_research::ChristofidesPathSolver::MatchingAlgorithm</a></div><div class="ttdeci">MatchingAlgorithm</div><div class="ttdef"><b>Definition:</b> <a href="christofides_8h_source.html#l00042">christofides.h:42</a></div></div>
<div class="ttc" id="anamespaceoperations__research_html_a272bcc4d6ee0e87a7fc430a5cb13c582"><div class="ttname"><a href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">operations_research::NodeIndex</a></div><div class="ttdeci">int32 NodeIndex</div><div class="ttdef"><b>Definition:</b> <a href="ebert__graph_8h_source.html#l00200">ebert_graph.h:192</a></div></div>
<div class="ttc" id="aclassoperations__research_1_1_christofides_path_solver_html_a1d4f082de5fc3eed348d65eb30b5f3e7a201b88f6589fa1271207fe29f583dc96"><div class="ttname"><a href="classoperations__research_1_1_christofides_path_solver.html#a1d4f082de5fc3eed348d65eb30b5f3e7a201b88f6589fa1271207fe29f583dc96">operations_research::ChristofidesPathSolver::MatchingAlgorithm::MINIMUM_WEIGHT_MATCHING_WITH_MIP</a></div><div class="ttdeci">@ MINIMUM_WEIGHT_MATCHING_WITH_MIP</div></div>
<div class="ttc" id="aclassoperations__research_1_1_christofides_path_solver_html_a3076d9001536ea98d419faa81e7d8a47"><div class="ttname"><a href="classoperations__research_1_1_christofides_path_solver.html#a3076d9001536ea98d419faa81e7d8a47">operations_research::ChristofidesPathSolver::TravelingSalesmanPath</a></div><div class="ttdeci">std::vector&lt; NodeIndex &gt; TravelingSalesmanPath()</div><div class="ttdef"><b>Definition:</b> <a href="christofides_8h_source.html#l00231">christofides.h:231</a></div></div>
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