274 lines
14 KiB
HTML
274 lines
14 KiB
HTML
<html><head><!--
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Copyright 2005 Aaron Windsor
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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: Aaron Windsor
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--><title>Boost Graph Library: Maximum Cardinality 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_cardinality_matching">Maximum Cardinality 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 edmonds_maximum_cardinality_matching(const Graph& g, MateMap mate);
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template <typename Graph, typename MateMap, typename VertexIndexMap>
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void edmonds_maximum_cardinality_matching(const Graph& g, MateMap mate, VertexIndexMap vm);
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template <typename Graph, typename MateMap>
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bool checked_edmonds_maximum_cardinality_matching(const Graph& g, MateMap mate);
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template <typename Graph, typename MateMap, typename VertexIndexMap>
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bool checked_edmonds_maximum_cardinality_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:matching">A <i>matching</i> is a subset of the edges
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of a graph such that no two edges share a common vertex.
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Two different matchings in the same graph are illustrated below (edges in the
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matching are colored blue.) The matching on the left is a <i>maximal matching</i>,
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meaning that its size can't be increased by adding edges. The matching on the
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right is a <i>maximum cardinality matching</i>, meaning that is has maximum size
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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_matching"><img src="figs/maximal-match.png"></a></td>
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<td width="150"></td>
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<td><a name="fig:maximum_matching"><img src="figs/maximum-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>edmonds_maximum_cardinality_matching</tt> and
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<tt>checked_edmonds_maximum_cardinality_matching</tt> find the
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maximum cardinality 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>. The only difference between
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<tt>edmonds_maximum_cardinality_matching</tt> and
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<tt>checked_edmonds_maximum_cardinality_matching</tt> is that as a final step,
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the latter algorithm runs a simple verification on the matching computed and
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returns <tt>true</tt> if and only if the matching is indeed
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a maximum cardinality matching.
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<p>
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Given a matching M, any vertex that isn't covered by an edge in M is called <i>free</i>. Any
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simple path containing exactly <i>2n + 1</i> edges that starts and ends at free vertices and contains
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<i>n</i> edges from M is called an <i>alternating path</i>. Given an alternating path <i>p</i>, all matching and
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non-matching edges on <i>p</i> can be swapped, resulting in a new matching that's larger than the
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original matching by exactly one edge. This method of incrementally increasing the size of matching, along
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with the following fact, forms the basis of Edmonds' matching algorithm:
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<blockquote>
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<i>An alternating path through the matching M exists if and only if M is not a maximum cardinality matching.</i>
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</blockquote>
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The difficult part is, of course, finding an augmenting path whenever one exists.
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The algorithm we use for finding a maximum cardinality matching consists of three basic steps:
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<ol>
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<li>Create an initial matching.
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<li>Repeatedly find an augmenting path and use it to increase the size of the matching until no augmenting path exists.
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<li>Verify that the matching found is a maximum cardinality matching.
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</ol>
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If you use <tt>checked_edmonds_maximum_cardinality_matching</tt> or
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<tt>edmonds_maximum_cardinality_matching</tt>, all three of these
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steps are chosen for you, but it's easy to plug in different algorithms for these three steps
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using a generic matching function discussed below - in fact, both <tt>checked_edmonds_maximum_cardinality_matching</tt>
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and <tt>edmonds_maximum_cardinality_matching</tt> are just inlined specializations of this function.
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<p>
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When quoting time bounds for algorithms, we assume that <tt>VertexIndexMap</tt> is a property map
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that allows for constant-time mapping between vertices and indices (which is easily achieved if,
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for instance, the vertices are stored in contiguous memory.) We use <i>n</i> and <i>m</i> to represent the size
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of the vertex and edge sets, respectively, of the input graph.
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<h4>Algorithms for Creating an Initial Matching</h4>
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<ul>
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<li><b><tt>empty_matching</tt></b>: Takes time <i>O(n)</i> to initialize the empty matching.
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<li><b><tt>greedy_matching</tt></b>: The matching obtained by iterating through the edges and adding an edge
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if it doesn't conflict with the edges already in the matching. This matching is maximal, and is therefore
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guaranteed to contain at least half of the edges that a maximum matching has. Takes time <i>O(m log n)</i>.
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<li><b><tt>extra_greedy_matching</tt></b>: Sorts the edges in increasing order of the degree of the vertices
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contained in each edge, then constructs a greedy matching from those edges. Also a maximal matching, and can
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sometimes be much closer to the maximum cardinality matching than a simple <tt>greedy_matching</tt>.
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Takes time <i>O(m log n)</i>, but the constants involved make this a slower algorithm than
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<tt>greedy_matching</tt>.
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</ul>
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<h4>Algorithms for Finding an Augmenting Path</h4>
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<ul>
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<li><b><tt>edmonds_augmenting_path_finder</tt></b>: Finds an augmenting path in time <i>O(m alpha(m,n))</i>,
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where <i>alpha(m,n)</i> is an inverse of the Ackerman function. <i>alpha(m,n)</i> is one of the slowest
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growing functions that occurs naturally in computer science; essentially, <i>alpha(m,n)</i> ≤ 4 for any
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graph that we'd ever hope to run this algorithm on. Since we arrive at a maximum cardinality matching after
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augmenting <i>O(n)</i> matchings, the entire algorithm takes time <i>O(mn alpha(m,n))</i>. Edmonds' original
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algorithm appeared in [<a href="bibliography.html#edmonds65">64</a>], but our implementation of
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Edmonds' algorithm closely follows Tarjan's
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description of the algorithm from [<a href="bibliography.html#tarjan83:_data_struct_network_algo">27</a>].
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<li><b><tt>no_augmenting_path_finder</tt></b>: Can be used if no augmentation of the initial matching is desired.
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</ul>
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<h4>Verification Algorithms</h4>
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<ul>
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<li><b><tt>maximum_cardinality_matching_verifier</tt></b>: Returns true if and only if the matching found is a
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maximum cardinality matching. Takes time <i>O(m alpha(m,n))</i>, which is on the order of a single iteration
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of Edmonds' algorithm.
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<li><b><tt>no_matching_verifier</tt></b>: Always returns true
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</ul>
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Why is a verification algorithm needed? Edmonds' algorithm is fairly complex, and it's nearly
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impossible for a human without a few days of spare time to figure out if the matching produced by
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<tt>edmonds_matching</tt> on a graph with, say, 100 vertices and 500 edges is indeed a maximum cardinality
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matching. A verification algorithm can do this mechanically, and it's much easier to verify by inspection
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that the verification algorithm has been implemented correctly than it is to verify by inspection that
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Edmonds' algorithm has been implemented correctly.
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The Boost Graph library makes it incredibly simple to perform the subroutines needed by the verifier
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(such as finding all the connected components of odd cardinality in a graph, or creating the induced graph
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on all vertices with a certain label) in just a few lines of code.
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<p>
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Understanding how the verifier works requires a few graph-theoretic facts.
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Let <i>m(G)</i> be the size of a maximum cardinality matching in the graph <i>G</i>.
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Denote by <i>o(G)</i> the number of connected components in <i>G</i> of odd cardinality, and for a set of
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vertices <i>X</i>, denote by <i>G - X</i> the induced graph on the vertex set <i>V(G) - X</i>. Then the
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Tutte-Berge Formula says that
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<blockquote>
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<i>2 * m(G) = min ( |V(G)| + |X| - o(G-X) )</i>
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</blockquote>
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Where the minimum is taken over all subsets <i>X</i> of the vertex set <i>V(G)</i>. A side effect of the
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Edmonds Blossom-Shrinking algorithm is that it computes what is known as the Edmonds-Gallai decomposition
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of a graph: it decomposes the graph into three disjoint sets of vertices, one of which achieves the minimum
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in the Tutte-Berge Formula.
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An outline of our verification procedure is:
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Given a <tt>Graph g</tt> and <tt>MateMap mate</tt>,
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<ol>
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<li>Check to make sure that <tt>mate</tt> is a valid matching on <tt>g</tt>.
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<li>Run <tt>edmonds_augmenting_path_finder</tt> once on <tt>g</tt> and <tt>mate</tt>. If it finds an augmenting
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path, the matching isn't a maximum cardinality matching. Otherwise, we retrieve a copy of the <tt>vertex_state</tt>
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map used by the <tt>edmonds_augmenting_path_finder</tt>. The Edmonds-Gallai decomposition tells us that the set
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of vertices labeled <tt>V_ODD</tt> by the <tt>vertex_state</tt> map can be used as the set X to achieve the
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minimum in the Tutte-Berge Formula.
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<li>Count the number of vertices labeled <tt>V_ODD</tt>, store this in <tt>num_odd_vertices</tt>.
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<li>Create a <a href="filtered_graph.html"><tt>filtered_graph</tt></a>
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consisting of all vertices that aren't labeled <tt>V_ODD</tt>. Count the number of odd connected components
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in this graph and store the result in <tt>num_odd_connected_components</tt>.
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<li>Test to see if equality holds in the Tutte-Berge formula using |X| = <tt>num_odd_vertices</tt> and o(G-X) = <tt>num_odd_connected_components</tt>. Return true if it holds, false otherwise.
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</ol>
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Assuming these steps are implemented correctly, the verifier will never return a false positive,
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and will only return a false negative if <tt>edmonds_augmenting_path_finder</tt> doesn't compute the
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<tt>vertex_state</tt> map correctly, in which case the <tt>edmonds_augmenting_path_finder</tt>
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isn't working correctly.
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<h4>Creating Your Own Matching Algorithms</h4>
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Creating a matching algorithm is as simple as plugging the algorithms described above into a generic
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matching function, which has the following signature:
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<pre>
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template <typename Graph,
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typename MateMap,
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typename VertexIndexMap,
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template <typename, typename, typename> class AugmentingPathFinder,
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template <typename, typename> class InitialMatchingFinder,
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template <typename, typename, typename> class MatchingVerifier>
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bool matching(const Graph& g, MateMap mate, VertexIndexMap vm)
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</pre>
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The matching functions provided for you are just inlined specializations of this function:
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<tt>edmonds_maximum_cardinality_matching</tt> uses <tt>edmonds_augmenting_path_finder</tt>
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as the <tt>AugmentingPathFinder</tt>, <tt>extra_greedy_matching</tt> as the <tt>InitialMatchingFinder</tt>,
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and <tt>no_matching_verifier</tt> as the <tt>MatchingVerifier</tt>.
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<tt>checked_edmonds_maximum_cardinality_matching</tt> uses the same parameters except that
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<tt>maximum_cardinality_matching_verifier</tt> is used for the <tt>MatchingVerifier</tt>.
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<p>
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These aren't necessarily the best choices for any situation - for example, it's been claimed in the literature
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that for sparse graphs, Edmonds' algorithm converges to the maximum cardinality matching more quickly if it
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isn't supplied with an intitial matching. Such an algorithm can be easily assembled by calling <tt>matching</tt> with
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<ul>
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<li><tt>AugmentingPathFinder = edmonds_augmenting_path_finder</tt>
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<li><tt>InitialMatchingFinder = empty_matching</tt>
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</ul>
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and choosing the <tt>MatchingVerifier</tt> depending on how careful you're feeling.
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<p>
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Suppose instead that you want a relatively large matching quickly, but are not exactly interested in a maximum matching.
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Both extra_greedy_matching and greedy_matching find maximal matchings, which means they're guaranteed to be at
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least half the size of a maximum cardinality matching, so you could call <tt>matching</tt> with
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<ul>
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<li><tt>AugmentingPathFinder = no_augmenting_path_finder</tt>
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<li><tt>InitialMatchingFinder = extra_greedy_matching</tt>
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<li><tt>MatchingVerifier = maximum_cardinality_matching_verifier</tt>
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</ul>
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The resulting algorithm will find an extra greedy matching in time <i>O(m log n)</i> without looking for
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augmenting paths. As a bonus, the return value of this function is true if and only if the extra greedy
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matching happens to be a maximum cardinality matching.
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</p><h3>Where Defined</h3>
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<p>
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<a href="../../../boost/graph/max_cardinality_matching.hpp"><tt>boost/graph/max_cardinality_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>.<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 lookups, the time complexity for both
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<tt>edmonds_matching</tt> and <tt>checked_edmonds_matching</tt> is <i>O(mn alpha(m,n))</i>.
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<i>alpha(m,n)</i> is a slow growing function that is at most 4 for any feasible input.
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</p><p>
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</p><h3>Example</h3>
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<p> The file <a href="../example/matching_example.cpp"><tt>example/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 <20> 2005</td><td>
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Aaron Windsor (<a href="mailto:aaron.windsor@gmail.com">aaron.windsor@gmail.com</a>)<br>
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</td></tr></tbody></table>
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</body></html>
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