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[CI SKIP] documentation for maximum weighted matching
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@ -148,7 +148,7 @@ J. W. H. Liu.
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<P></P><DT><A NAME="mehlhorn99:_leda">22</A>
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<DD>
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K. Mehlhorn and S. Näher.
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K. Mehlhorn and S. Näher.
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<BR><EM>The LEDA Platform of Combinatorial and Geometric Computing</EM>.
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<BR>Cambridge University Press, 1999.
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@ -438,6 +438,16 @@ David Bruce Wilson
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<BR><em>Generating random spanning trees more quickly than the cover time</em>.
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ACM Symposium on the Theory of Computing, pp. 296-303, 1996.
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<p></p><dt><a name="edmonds65:_max_weighted_match">74</a>
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<dd>J. Edmonds<br>
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<em>Maximum Matching and a Polyhedron with 0, 1-Vertices</em><br>
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Journal of Research of the National Bureau of Standards B 69, pp. 125-130, 1965.
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<p></p><dt><a name="gabow90">75</a>
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<dd>Harold N. Gabow<br>
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<em>Data Structures for Weighted Matching and Nearest Common Ancestors with Linking</em><br>
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Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 434-443, 1990.
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</dl>
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<br>
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<html><head><!--
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Copyright 2018 Yi Ji
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Use, modification and distribution is subject to the Boost Software
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License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt)
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Author: Yi Ji
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--><title>Boost Graph Library: Maximum Weighted Matching</title></head>
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<body alink="#ff0000" bgcolor="#ffffff" link="#0000ee" text="#000000" vlink="#551a8b">
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<img src="../../../boost.png" alt="C++ Boost" height="86" width="277">
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<br clear="">
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<h1>
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<a name="sec:maximum_weighted_matching">Maximum Weighted Matching</a>
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</h1>
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<pre>
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template <typename Graph, typename MateMap>
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void maximum_weighted_matching(const Graph& g, MateMap mate);
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template <typename Graph, typename MateMap, typename VertexIndexMap>
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void maximum_weighted_matching(const Graph& g, MateMap mate, VertexIndexMap vm);
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template <typename Graph, typename MateMap>
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void brute_force_maximum_weighted_matching(const Graph& g, MateMap mate);
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template <typename Graph, typename MateMap, typename VertexIndexMap>
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void brute_force_maximum_weighted_matching(const Graph& g, MateMap mate, VertexIndexMap vm);
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</pre>
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<p>
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<a name="sec:weighted_matching">Before you continue, it is recommended to read
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about <a href="./maximum_matching.html">maximal cardinality matching</a> first.
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A <i>maximum weighted matching</i> of an edge-weighted graph is a matching
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for which the sum of the weights of the edges is maximum.
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Two different matchings (edges in the matching are colored blue) in the same graph are illustrated below.
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The matching on the left is a maximum cardinality matching of size 8 and a maximal
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weighted matching of weight sum 30, meaning that is has maximum size over all matchings in the graph
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and its weight sum can't be increased by adding edges.
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The matching on the right is a maximum weighted matching of size 7 and weight sum 38, meaning that it has maximum
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weight sum over all matchings in the graph.
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</a></p><p></p><center>
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<table border="0">
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<tr>
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<td><a name="fig:maximal_weighted_matching"><img src="figs/maximal-weighted-match.png"></a></td>
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<td width="150"></td>
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<td><a name="fig:maximum_weighted_matching"><img src="figs/maximum-weighted-match.png"></a></td>
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</tr>
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</table>
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</center>
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<p>
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Both <tt>maximum_weighted_matching</tt> and
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<tt>brute_force_maximum_weighted_matching</tt> find a
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maximum weighted matching in any undirected graph. The matching is returned in a
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<tt>MateMap</tt>, which is a
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<a href="../../property_map/doc/ReadWritePropertyMap.html">ReadWritePropertyMap</a>
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that maps vertices to vertices. In the mapping returned, each vertex is either mapped
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to the vertex it's matched to, or to <tt>graph_traits<Graph>::null_vertex()</tt> if it
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doesn't participate in the matching. If no <tt>VertexIndexMap</tt> is provided, both functions
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assume that the <tt>VertexIndexMap</tt> is provided as an internal graph property accessible
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by calling <tt>get(vertex_index, g)</tt>.
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<p>
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The maximum weighted matching problem was solved by Edmonds in [<a href="bibliography.html#edmonds65:_max_weighted_match">74</a>].
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The implementation of <tt>maximum_weighted_matching</tt> followed Chapter 6, Section 10 of [<a href="bibliography.html#lawler76:_comb_opt">20</a>] and
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was written in a consistent style with <tt>edmonds_maximum_cardinality_matching</tt> because of their algorithmic similarity.
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In addition, a brute-force verifier <tt>brute_force_maximum_weighted_matching</tt> simply searches all possible matchings in any graph and selects one with the maximum weight sum.
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</p><h3>Algorithm Description</h3>
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Primal-dual method in linear programming is introduced to solve weighted matching problems. Edmonds proved that for any graph,
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the maximum number of edges in a matching is equal to the minimum capacity of an odd-set cover; this further enable us to prove a max-min duality theorem for weighted matching.
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Let <i>H<sub>k-1</sub></i> denote any graph obtained from G by contracting odd sets of three or more nodes and deleting single nodes,
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where the capacity of the family of odd sets (not necessarily a cover of G) is <i>k-1</i>. Let <i>X<sub>k</sub></i> denote any matching containing <i>k</i> edges.
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Each edge <i>(i, j)</i> has a weight <i>w<sub>ij</sub></i>. We have:
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<math>
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max<i><sub>X<sub>k</sub></sub></i> min {<i>w<sub>ij</sub>|(i, j) ϵ X<sub>k</sub></i>} = min<i><sub>H<sub>k-1</sub></sub></i> max {<i>w<sub>ij</sub>|(i, j) ϵ H<sub>k-1</sub></i>}.
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</math>
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This matching duality theorem gives an indication of how the matching problem should be formulated as a linear programming problem. That is,
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the theorem suggests a set of linear inequalities which are satisfied by any matching, and it is anticipated that these inequalities describe a convex polyhedron
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with integer vertices corresponding to feasible matchings.
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<p>
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For <tt>maximum_weighted_matching</tt>, the management of blossoms is much more involved than in the case of <tt>max_cardinality_matching</tt>.
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It is not sufficient to record only the outermost blossoms. When an outermost blossom is expanded,
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it is necessary to know which blossom are nested immediately with it, so that these blossoms can be restored to the status of the outermost blossoms.
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When augmentation occurs, blossoms with strictly positive dual variables must be maintained for use in the next application of the labeling procedure.
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<p>
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The outline of the algorithm is as follow:
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<ol start="0">
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<li>Start with an empty matching and initialize dual variables as a half of maximum edge weight.</li>
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<li>(Labeling) Root an alternate tree at each exposed node, and proceed to construct alternate trees by labeling, using only edges with zero slack value.
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If an augmenting path is found, go to step 2. If a blossom is formed, go to step 3. Otherwise, go to step 4. </li>
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<li>(Augmentation) Find the augmenting path, tracing the path through shrunken blossoms. Augment the matching,
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correct labels on nodes in the augmenting path, expand blossoms with zero dual variables and remove labels from all base nodes. Go to step 1.</li>
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<li>(Blossoming) Determine the membership and base node of the new blossom and supply missing labels for all non-base nodes in the blossom.
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Return to step 1.</li>
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<li>(Revision of Dual Solution) Adjust the dual variables based on the primal-dual method. Go to step 1 or halt, accordingly.</li>
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</ol>
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Note that in <tt>maximum_weighted_matching</tt>, all edge weights are multiplied by 4, so that all dual variables always remain as integers if all edge weights are integers.
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Unlike <tt>max_cardinality_matching</tt>, the initial matching and augmenting path finder are not parameterized,
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because the algorithm maintains blossoms, dual variables and node labels across all augmentations.
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The algorithm's time complexity is reduced from <i>O(V<sup>4</sup>)</i> (naive implementation of [<a href="bibliography.html#edmonds65:_max_weighted_match">74</a>])
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to <i>O(V<sup>3</sup>)</i>, by the avoidance of re-scanning labels after revision of the dual solution.
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Several special node variables <i>pi, tau, gamma</i> and arrays <i>critical_edge, tau_idx</i> are introduced for this purpose.
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Please refer to [<a href="bibliography.html#lawler76:_comb_opt">20</a>] and code comments for more implementation details.
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</p><h3>Where Defined</h3>
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<p>
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<a href="../../../boost/graph/maximum_weighted_matching.hpp"><tt>boost/graph/maximum_weighted_matching.hpp</tt></a>
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</p><h3>Parameters</h3>
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IN: <tt>const Graph& g</tt>
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<blockquote>
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An undirected graph. The graph type must be a model of
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<a href="VertexAndEdgeListGraph.html">Vertex and Edge List Graph</a> and
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<a href="IncidenceGraph.html">Incidence Graph</a>.
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The edge property of the graph <tt>property_map<Graph, edge_weight_t></tt> must exist and have numeric value type.<br>
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</blockquote>
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IN: <tt>VertexIndexMap vm</tt>
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<blockquote>
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Must be a model of <a href="../../property_map/doc/ReadablePropertyMap.html">ReadablePropertyMap</a>, mapping vertices to integer indices.
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</blockquote>
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OUT: <tt>MateMap mate</tt>
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<blockquote>
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Must be a model of <a href="../../property_map/doc/ReadWritePropertyMap.html">ReadWritePropertyMap</a>, mapping
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vertices to vertices. For any vertex v in the graph, <tt>get(mate,v)</tt> will be the vertex that v is matched to, or
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<tt>graph_traits<Graph>::null_vertex()</tt> if v isn't matched.
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</blockquote>
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<h3>Complexity</h3>
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<p>
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Let <i>m</i> and <i>n</i> be the number of edges and vertices in the input graph, respectively. Assuming the
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<tt>VertexIndexMap</tt> supplied allows constant-time lookup, the time complexity for
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<tt>maximum_weighted_matching</tt> is <i>O(n<sup>3</sup>)</i>. For <tt>brute_force_maximum_weighted_matching</tt>, the time complexity is exponential of <i>m</i>.
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Note that the best known time complexity for maximum weighted matching in general graph
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is <i>O(nm+n<sup>2</sup>log(n))</i> by [<a href="bibliography.html#gabow90">75</a>], but relies on an
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efficient algorithm for solving nearest ancestor problem on trees, which is not provided in Boost C++ libraries.
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</p><p>
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</p><h3>Example</h3>
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<p> The file <a href="../example/weighted_matching_example.cpp"><tt>example/weighted_matching_example.cpp</tt></a>
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contains an example.
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<br>
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</p><hr>
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<table>
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<tbody><tr valign="top">
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<td nowrap="nowrap">Copyright © 2018</td><td>
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Yi Ji (<a href="mailto:jiy@pku.edu.cn">jiy@pku.edu.cn</a>)<br>
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</td></tr></tbody></table>
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</body></html>
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