| <html><head><!-- |
| Copyright 2005 Aaron Windsor |
| |
| Use, modification and distribution is subject to the Boost Software |
| License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at |
| http://www.boost.org/LICENSE_1_0.txt) |
| |
| Author: Aaron Windsor |
| --><title>Boost Graph Library: Maximum Cardinality Matching</title></head> |
| <body alink="#ff0000" bgcolor="#ffffff" link="#0000ee" text="#000000" vlink="#551a8b"> |
| <img src="../../../boost.png" alt="C++ Boost" height="86" width="277"> |
| <br clear=""> |
| <h1> |
| <a name="sec:maximum_cardinality_matching">Maximum Cardinality Matching</a> |
| </h1> |
| <pre> |
| template <typename Graph, typename MateMap> |
| void edmonds_maximum_cardinality_matching(const Graph& g, MateMap mate); |
| |
| template <typename Graph, typename MateMap, typename VertexIndexMap> |
| void edmonds_maximum_cardinality_matching(const Graph& g, MateMap mate, VertexIndexMap vm); |
| |
| template <typename Graph, typename MateMap> |
| bool checked_edmonds_maximum_cardinality_matching(const Graph& g, MateMap mate); |
| |
| template <typename Graph, typename MateMap, typename VertexIndexMap> |
| bool checked_edmonds_maximum_cardinality_matching(const Graph& g, MateMap mate, VertexIndexMap vm); |
| </pre> |
| <p> |
| <a name="sec:matching">A <i>matching</i> is a subset of the edges |
| of a graph such that no two edges share a common vertex. |
| Two different matchings in the same graph are illustrated below (edges in the |
| matching are colored blue.) The matching on the left is a <i>maximal matching</i>, |
| meaning that its size can't be increased by adding edges. The matching on the |
| right is a <i>maximum cardinality matching</i>, meaning that is has maximum size |
| over all matchings in the graph. |
| |
| </a></p><p></p><center> |
| <table border="0"> |
| <tr> |
| <td><a name="fig:maximal_matching"><img src="figs/maximal-match.png"></a></td> |
| <td width="150"></td> |
| <td><a name="fig:maximum_matching"><img src="figs/maximum-match.png"></a></td> |
| </tr> |
| </table> |
| </center> |
| |
| <p> |
| Both <tt>edmonds_maximum_cardinality_matching</tt> and |
| <tt>checked_edmonds_maximum_cardinality_matching</tt> find the |
| maximum cardinality matching in any undirected graph. The matching is returned in a |
| <tt>MateMap</tt>, which is a |
| <a href="../../property_map/doc/ReadWritePropertyMap.html">ReadWritePropertyMap</a> |
| that maps vertices to vertices. In the mapping returned, each vertex is either mapped |
| to the vertex it's matched to, or to <tt>graph_traits<Graph>::null_vertex()</tt> if it |
| doesn't participate in the matching. If no <tt>VertexIndexMap</tt> is provided, both functions |
| assume that the <tt>VertexIndexMap</tt> is provided as an internal graph property accessible |
| by calling <tt>get(vertex_index,g)</tt>. The only difference between |
| <tt>edmonds_maximum_cardinality_matching</tt> and |
| <tt>checked_edmonds_maximum_cardinality_matching</tt> is that as a final step, |
| the latter algorithm runs a simple verification on the matching computed and |
| returns <tt>true</tt> if and only if the matching is indeed |
| a maximum cardinality matching. |
| |
| <p> |
| Given a matching M, any vertex that isn't covered by an edge in M is called <i>free</i>. Any |
| simple path containing exactly <i>2n + 1</i> edges that starts and ends at free vertices and contains |
| <i>n</i> edges from M is called an <i>alternating path</i>. Given an alternating path <i>p</i>, all matching and |
| non-matching edges on <i>p</i> can be swapped, resulting in a new matching that's larger than the |
| original matching by exactly one edge. This method of incrementally increasing the size of matching, along |
| with the following fact, forms the basis of Edmonds' matching algorithm: |
| |
| <blockquote> |
| <i>An alternating path through the matching M exists if and only if M is not a maximum cardinality matching.</i> |
| </blockquote> |
| |
| The difficult part is, of course, finding an augmenting path whenever one exists. |
| The algorithm we use for finding a maximum cardinality matching consists of three basic steps: |
| <ol> |
| <li>Create an initial matching. |
| <li>Repeatedly find an augmenting path and use it to increase the size of the matching until no augmenting path exists. |
| <li>Verify that the matching found is a maximum cardinality matching. |
| </ol> |
| |
| If you use <tt>checked_edmonds_maximum_cardinality_matching</tt> or |
| <tt>edmonds_maximum_cardinality_matching</tt>, all three of these |
| steps are chosen for you, but it's easy to plug in different algorithms for these three steps |
| using a generic matching function discussed below - in fact, both <tt>checked_edmonds_maximum_cardinality_matching</tt> |
| and <tt>edmonds_maximum_cardinality_matching</tt> are just inlined specializations of this function. |
| |
| <p> |
| When quoting time bounds for algorithms, we assume that <tt>VertexIndexMap</tt> is a property map |
| that allows for constant-time mapping between vertices and indices (which is easily achieved if, |
| for instance, the vertices are stored in contiguous memory.) We use <i>n</i> and <i>m</i> to represent the size |
| of the vertex and edge sets, respectively, of the input graph. |
| |
| <h4>Algorithms for Creating an Initial Matching</h4> |
| |
| <ul> |
| <li><b><tt>empty_matching</tt></b>: Takes time <i>O(n)</i> to initialize the empty matching. |
| <li><b><tt>greedy_matching</tt></b>: The matching obtained by iterating through the edges and adding an edge |
| if it doesn't conflict with the edges already in the matching. This matching is maximal, and is therefore |
| guaranteed to contain at least half of the edges that a maximum matching has. Takes time <i>O(m log n)</i>. |
| <li><b><tt>extra_greedy_matching</tt></b>: Sorts the edges in increasing order of the degree of the vertices |
| contained in each edge, then constructs a greedy matching from those edges. Also a maximal matching, and can |
| sometimes be much closer to the maximum cardinality matching than a simple <tt>greedy_matching</tt>. |
| Takes time <i>O(m log n)</i>, but the constants involved make this a slower algorithm than |
| <tt>greedy_matching</tt>. |
| </ul> |
| |
| <h4>Algorithms for Finding an Augmenting Path</h4> |
| |
| <ul> |
| <li><b><tt>edmonds_augmenting_path_finder</tt></b>: Finds an augmenting path in time <i>O(m alpha(m,n))</i>, |
| where <i>alpha(m,n)</i> is an inverse of the Ackerman function. <i>alpha(m,n)</i> is one of the slowest |
| growing functions that occurs naturally in computer science; essentially, <i>alpha(m,n)</i> ≤ 4 for any |
| graph that we'd ever hope to run this algorithm on. Since we arrive at a maximum cardinality matching after |
| augmenting <i>O(n)</i> matchings, the entire algorithm takes time <i>O(mn alpha(m,n))</i>. Edmonds' original |
| algorithm appeared in [<a href="bibliography.html#edmonds65">64</a>], but our implementation of |
| Edmonds' algorithm closely follows Tarjan's |
| description of the algorithm from [<a href="bibliography.html#tarjan83:_data_struct_network_algo">27</a>]. |
| <li><b><tt>no_augmenting_path_finder</tt></b>: Can be used if no augmentation of the initial matching is desired. |
| </ul> |
| |
| <h4>Verification Algorithms</h4> |
| |
| <ul> |
| <li><b><tt>maximum_cardinality_matching_verifier</tt></b>: Returns true if and only if the matching found is a |
| maximum cardinality matching. Takes time <i>O(m alpha(m,n))</i>, which is on the order of a single iteration |
| of Edmonds' algorithm. |
| <li><b><tt>no_matching_verifier</tt></b>: Always returns true |
| </ul> |
| |
| Why is a verification algorithm needed? Edmonds' algorithm is fairly complex, and it's nearly |
| impossible for a human without a few days of spare time to figure out if the matching produced by |
| <tt>edmonds_matching</tt> on a graph with, say, 100 vertices and 500 edges is indeed a maximum cardinality |
| matching. A verification algorithm can do this mechanically, and it's much easier to verify by inspection |
| that the verification algorithm has been implemented correctly than it is to verify by inspection that |
| Edmonds' algorithm has been implemented correctly. |
| The Boost Graph library makes it incredibly simple to perform the subroutines needed by the verifier |
| (such as finding all the connected components of odd cardinality in a graph, or creating the induced graph |
| on all vertices with a certain label) in just a few lines of code. |
| |
| <p> |
| Understanding how the verifier works requires a few graph-theoretic facts. |
| Let <i>m(G)</i> be the size of a maximum cardinality matching in the graph <i>G</i>. |
| Denote by <i>o(G)</i> the number of connected components in <i>G</i> of odd cardinality, and for a set of |
| vertices <i>X</i>, denote by <i>G - X</i> the induced graph on the vertex set <i>V(G) - X</i>. Then the |
| Tutte-Berge Formula says that |
| <blockquote> |
| <i>2 * m(G) = min ( |V(G)| + |X| - o(G-X) )</i> |
| </blockquote> |
| Where the minimum is taken over all subsets <i>X</i> of the vertex set <i>V(G)</i>. A side effect of the |
| Edmonds Blossom-Shrinking algorithm is that it computes what is known as the Edmonds-Gallai decomposition |
| of a graph: it decomposes the graph into three disjoint sets of vertices, one of which achieves the minimum |
| in the Tutte-Berge Formula. |
| |
| An outline of our verification procedure is: |
| |
| Given a <tt>Graph g</tt> and <tt>MateMap mate</tt>, |
| <ol> |
| <li>Check to make sure that <tt>mate</tt> is a valid matching on <tt>g</tt>. |
| <li>Run <tt>edmonds_augmenting_path_finder</tt> once on <tt>g</tt> and <tt>mate</tt>. If it finds an augmenting |
| path, the matching isn't a maximum cardinality matching. Otherwise, we retrieve a copy of the <tt>vertex_state</tt> |
| map used by the <tt>edmonds_augmenting_path_finder</tt>. The Edmonds-Gallai decomposition tells us that the set |
| of vertices labeled <tt>V_ODD</tt> by the <tt>vertex_state</tt> map can be used as the set X to achieve the |
| minimum in the Tutte-Berge Formula. |
| <li>Count the number of vertices labeled <tt>V_ODD</tt>, store this in <tt>num_odd_vertices</tt>. |
| <li>Create a <a href="filtered_graph.html"><tt>filtered_graph</tt></a> |
| consisting of all vertices that aren't labeled <tt>V_ODD</tt>. Count the number of odd connected components |
| in this graph and store the result in <tt>num_odd_connected_components</tt>. |
| <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. |
| </ol> |
| |
| Assuming these steps are implemented correctly, the verifier will never return a false positive, |
| and will only return a false negative if <tt>edmonds_augmenting_path_finder</tt> doesn't compute the |
| <tt>vertex_state</tt> map correctly, in which case the <tt>edmonds_augmenting_path_finder</tt> |
| isn't working correctly. |
| |
| |
| <h4>Creating Your Own Matching Algorithms</h4> |
| |
| Creating a matching algorithm is as simple as plugging the algorithms described above into a generic |
| matching function, which has the following signature: |
| <pre> |
| template <typename Graph, |
| typename MateMap, |
| typename VertexIndexMap, |
| template <typename, typename, typename> class AugmentingPathFinder, |
| template <typename, typename> class InitialMatchingFinder, |
| template <typename, typename, typename> class MatchingVerifier> |
| bool matching(const Graph& g, MateMap mate, VertexIndexMap vm) |
| </pre> |
| The matching functions provided for you are just inlined specializations of this function: |
| <tt>edmonds_maximum_cardinality_matching</tt> uses <tt>edmonds_augmenting_path_finder</tt> |
| as the <tt>AugmentingPathFinder</tt>, <tt>extra_greedy_matching</tt> as the <tt>InitialMatchingFinder</tt>, |
| and <tt>no_matching_verifier</tt> as the <tt>MatchingVerifier</tt>. |
| <tt>checked_edmonds_maximum_cardinality_matching</tt> uses the same parameters except that |
| <tt>maximum_cardinality_matching_verifier</tt> is used for the <tt>MatchingVerifier</tt>. |
| |
| <p> |
| These aren't necessarily the best choices for any situation - for example, it's been claimed in the literature |
| that for sparse graphs, Edmonds' algorithm converges to the maximum cardinality matching more quickly if it |
| isn't supplied with an intitial matching. Such an algorithm can be easily assembled by calling <tt>matching</tt> with |
| <ul> |
| <li><tt>AugmentingPathFinder = edmonds_augmenting_path_finder</tt> |
| <li><tt>InitialMatchingFinder = empty_matching</tt> |
| </ul> |
| and choosing the <tt>MatchingVerifier</tt> depending on how careful you're feeling. |
| |
| <p> |
| Suppose instead that you want a relatively large matching quickly, but are not exactly interested in a maximum matching. |
| Both extra_greedy_matching and greedy_matching find maximal matchings, which means they're guaranteed to be at |
| least half the size of a maximum cardinality matching, so you could call <tt>matching</tt> with |
| <ul> |
| <li><tt>AugmentingPathFinder = no_augmenting_path_finder</tt> |
| <li><tt>InitialMatchingFinder = extra_greedy_matching</tt> |
| <li><tt>MatchingVerifier = maximum_cardinality_matching_verifier</tt> |
| </ul> |
| The resulting algorithm will find an extra greedy matching in time <i>O(m log n)</i> without looking for |
| augmenting paths. As a bonus, the return value of this function is true if and only if the extra greedy |
| matching happens to be a maximum cardinality matching. |
| |
| </p><h3>Where Defined</h3> |
| |
| <p> |
| <a href="../../../boost/graph/max_cardinality_matching.hpp"><tt>boost/graph/max_cardinality_matching.hpp</tt></a> |
| |
| |
| </p><h3>Parameters</h3> |
| |
| IN: <tt>const Graph& g</tt> |
| <blockquote> |
| An undirected graph. The graph type must be a model of |
| <a href="VertexAndEdgeListGraph.html">Vertex and Edge List Graph</a> and |
| <a href="IncidenceGraph.html">Incidence Graph</a>.<br> |
| </blockquote> |
| |
| IN: <tt>VertexIndexMap vm</tt> |
| <blockquote> |
| Must be a model of <a href="../../property_map/doc/ReadablePropertyMap.html">ReadablePropertyMap</a>, mapping vertices to integer indices. |
| </blockquote> |
| |
| OUT: <tt>MateMap mate</tt> |
| <blockquote> |
| Must be a model of <a href="../../property_map/doc/ReadWritePropertyMap.html">ReadWritePropertyMap</a>, mapping |
| 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 |
| <tt>graph_traits<Graph>::null_vertex()</tt> if v isn't matched. |
| </blockquote> |
| |
| <h3>Complexity</h3> |
| |
| <p> |
| Let <i>m</i> and <i>n</i> be the number of edges and vertices in the input graph, respectively. Assuming the |
| <tt>VertexIndexMap</tt> supplied allows constant-time lookups, the time complexity for both |
| <tt>edmonds_matching</tt> and <tt>checked_edmonds_matching</tt> is <i>O(mn alpha(m,n))</i>. |
| <i>alpha(m,n)</i> is a slow growing function that is at most 4 for any feasible input. |
| </p><p> |
| |
| </p><h3>Example</h3> |
| |
| <p> The file <a href="../example/matching_example.cpp"><tt>example/matching_example.cpp</tt></a> |
| contains an example. |
| |
| <br> |
| </p><hr> |
| <table> |
| <tbody><tr valign="top"> |
| <td nowrap="nowrap">Copyright © 2005</td><td> |
| Aaron Windsor (<a href="mailto:aaron.windsor@gmail.com">aaron.windsor@gmail.com</a>)<br> |
| </td></tr></tbody></table> |
| |
| </body></html> |