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1 | <html><head><!-- |
2 | Copyright 2005 Aaron Windsor | |
3 | ||
4 | Use, modification and distribution is subject to the Boost Software | |
5 | License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at | |
6 | http://www.boost.org/LICENSE_1_0.txt) | |
7 | ||
8 | Author: Aaron Windsor | |
9 | --><title>Boost Graph Library: Maximum Cardinality Matching</title></head> | |
10 | <body alink="#ff0000" bgcolor="#ffffff" link="#0000ee" text="#000000" vlink="#551a8b"> | |
11 | <img src="../../../boost.png" alt="C++ Boost" height="86" width="277"> | |
12 | <br clear=""> | |
13 | <h1> | |
14 | <a name="sec:maximum_cardinality_matching">Maximum Cardinality Matching</a> | |
15 | </h1> | |
16 | <pre> | |
17 | template <typename Graph, typename MateMap> | |
18 | void edmonds_maximum_cardinality_matching(const Graph& g, MateMap mate); | |
19 | ||
20 | template <typename Graph, typename MateMap, typename VertexIndexMap> | |
21 | void edmonds_maximum_cardinality_matching(const Graph& g, MateMap mate, VertexIndexMap vm); | |
22 | ||
23 | template <typename Graph, typename MateMap> | |
24 | bool checked_edmonds_maximum_cardinality_matching(const Graph& g, MateMap mate); | |
25 | ||
26 | template <typename Graph, typename MateMap, typename VertexIndexMap> | |
27 | bool checked_edmonds_maximum_cardinality_matching(const Graph& g, MateMap mate, VertexIndexMap vm); | |
28 | </pre> | |
29 | <p> | |
30 | <a name="sec:matching">A <i>matching</i> is a subset of the edges | |
31 | of a graph such that no two edges share a common vertex. | |
32 | Two different matchings in the same graph are illustrated below (edges in the | |
33 | matching are colored blue.) The matching on the left is a <i>maximal matching</i>, | |
34 | meaning that its size can't be increased by adding edges. The matching on the | |
35 | right is a <i>maximum cardinality matching</i>, meaning that is has maximum size | |
36 | over all matchings in the graph. | |
37 | ||
38 | </a></p><p></p><center> | |
39 | <table border="0"> | |
40 | <tr> | |
41 | <td><a name="fig:maximal_matching"><img src="figs/maximal-match.png"></a></td> | |
42 | <td width="150"></td> | |
43 | <td><a name="fig:maximum_matching"><img src="figs/maximum-match.png"></a></td> | |
44 | </tr> | |
45 | </table> | |
46 | </center> | |
47 | ||
48 | <p> | |
49 | Both <tt>edmonds_maximum_cardinality_matching</tt> and | |
50 | <tt>checked_edmonds_maximum_cardinality_matching</tt> find the | |
51 | maximum cardinality matching in any undirected graph. The matching is returned in a | |
52 | <tt>MateMap</tt>, which is a | |
53 | <a href="../../property_map/doc/ReadWritePropertyMap.html">ReadWritePropertyMap</a> | |
54 | that maps vertices to vertices. In the mapping returned, each vertex is either mapped | |
55 | to the vertex it's matched to, or to <tt>graph_traits<Graph>::null_vertex()</tt> if it | |
56 | doesn't participate in the matching. If no <tt>VertexIndexMap</tt> is provided, both functions | |
57 | assume that the <tt>VertexIndexMap</tt> is provided as an internal graph property accessible | |
58 | by calling <tt>get(vertex_index,g)</tt>. The only difference between | |
59 | <tt>edmonds_maximum_cardinality_matching</tt> and | |
60 | <tt>checked_edmonds_maximum_cardinality_matching</tt> is that as a final step, | |
61 | the latter algorithm runs a simple verification on the matching computed and | |
62 | returns <tt>true</tt> if and only if the matching is indeed | |
63 | a maximum cardinality matching. | |
64 | ||
65 | <p> | |
66 | Given a matching M, any vertex that isn't covered by an edge in M is called <i>free</i>. Any | |
67 | simple path containing exactly <i>2n + 1</i> edges that starts and ends at free vertices and contains | |
68 | <i>n</i> edges from M is called an <i>alternating path</i>. Given an alternating path <i>p</i>, all matching and | |
69 | non-matching edges on <i>p</i> can be swapped, resulting in a new matching that's larger than the | |
70 | original matching by exactly one edge. This method of incrementally increasing the size of matching, along | |
71 | with the following fact, forms the basis of Edmonds' matching algorithm: | |
72 | ||
73 | <blockquote> | |
74 | <i>An alternating path through the matching M exists if and only if M is not a maximum cardinality matching.</i> | |
75 | </blockquote> | |
76 | ||
77 | The difficult part is, of course, finding an augmenting path whenever one exists. | |
78 | The algorithm we use for finding a maximum cardinality matching consists of three basic steps: | |
79 | <ol> | |
80 | <li>Create an initial matching. | |
81 | <li>Repeatedly find an augmenting path and use it to increase the size of the matching until no augmenting path exists. | |
82 | <li>Verify that the matching found is a maximum cardinality matching. | |
83 | </ol> | |
84 | ||
85 | If you use <tt>checked_edmonds_maximum_cardinality_matching</tt> or | |
86 | <tt>edmonds_maximum_cardinality_matching</tt>, all three of these | |
87 | steps are chosen for you, but it's easy to plug in different algorithms for these three steps | |
88 | using a generic matching function discussed below - in fact, both <tt>checked_edmonds_maximum_cardinality_matching</tt> | |
89 | and <tt>edmonds_maximum_cardinality_matching</tt> are just inlined specializations of this function. | |
90 | ||
91 | <p> | |
92 | When quoting time bounds for algorithms, we assume that <tt>VertexIndexMap</tt> is a property map | |
93 | that allows for constant-time mapping between vertices and indices (which is easily achieved if, | |
94 | for instance, the vertices are stored in contiguous memory.) We use <i>n</i> and <i>m</i> to represent the size | |
95 | of the vertex and edge sets, respectively, of the input graph. | |
96 | ||
97 | <h4>Algorithms for Creating an Initial Matching</h4> | |
98 | ||
99 | <ul> | |
100 | <li><b><tt>empty_matching</tt></b>: Takes time <i>O(n)</i> to initialize the empty matching. | |
101 | <li><b><tt>greedy_matching</tt></b>: The matching obtained by iterating through the edges and adding an edge | |
102 | if it doesn't conflict with the edges already in the matching. This matching is maximal, and is therefore | |
103 | guaranteed to contain at least half of the edges that a maximum matching has. Takes time <i>O(m log n)</i>. | |
104 | <li><b><tt>extra_greedy_matching</tt></b>: Sorts the edges in increasing order of the degree of the vertices | |
105 | contained in each edge, then constructs a greedy matching from those edges. Also a maximal matching, and can | |
106 | sometimes be much closer to the maximum cardinality matching than a simple <tt>greedy_matching</tt>. | |
107 | Takes time <i>O(m log n)</i>, but the constants involved make this a slower algorithm than | |
108 | <tt>greedy_matching</tt>. | |
109 | </ul> | |
110 | ||
111 | <h4>Algorithms for Finding an Augmenting Path</h4> | |
112 | ||
113 | <ul> | |
114 | <li><b><tt>edmonds_augmenting_path_finder</tt></b>: Finds an augmenting path in time <i>O(m alpha(m,n))</i>, | |
115 | where <i>alpha(m,n)</i> is an inverse of the Ackerman function. <i>alpha(m,n)</i> is one of the slowest | |
116 | growing functions that occurs naturally in computer science; essentially, <i>alpha(m,n)</i> ≤ 4 for any | |
117 | graph that we'd ever hope to run this algorithm on. Since we arrive at a maximum cardinality matching after | |
118 | augmenting <i>O(n)</i> matchings, the entire algorithm takes time <i>O(mn alpha(m,n))</i>. Edmonds' original | |
119 | algorithm appeared in [<a href="bibliography.html#edmonds65">64</a>], but our implementation of | |
120 | Edmonds' algorithm closely follows Tarjan's | |
121 | description of the algorithm from [<a href="bibliography.html#tarjan83:_data_struct_network_algo">27</a>]. | |
122 | <li><b><tt>no_augmenting_path_finder</tt></b>: Can be used if no augmentation of the initial matching is desired. | |
123 | </ul> | |
124 | ||
125 | <h4>Verification Algorithms</h4> | |
126 | ||
127 | <ul> | |
128 | <li><b><tt>maximum_cardinality_matching_verifier</tt></b>: Returns true if and only if the matching found is a | |
129 | maximum cardinality matching. Takes time <i>O(m alpha(m,n))</i>, which is on the order of a single iteration | |
130 | of Edmonds' algorithm. | |
131 | <li><b><tt>no_matching_verifier</tt></b>: Always returns true | |
132 | </ul> | |
133 | ||
134 | Why is a verification algorithm needed? Edmonds' algorithm is fairly complex, and it's nearly | |
135 | impossible for a human without a few days of spare time to figure out if the matching produced by | |
136 | <tt>edmonds_matching</tt> on a graph with, say, 100 vertices and 500 edges is indeed a maximum cardinality | |
137 | matching. A verification algorithm can do this mechanically, and it's much easier to verify by inspection | |
138 | that the verification algorithm has been implemented correctly than it is to verify by inspection that | |
139 | Edmonds' algorithm has been implemented correctly. | |
140 | The Boost Graph library makes it incredibly simple to perform the subroutines needed by the verifier | |
141 | (such as finding all the connected components of odd cardinality in a graph, or creating the induced graph | |
142 | on all vertices with a certain label) in just a few lines of code. | |
143 | ||
144 | <p> | |
145 | Understanding how the verifier works requires a few graph-theoretic facts. | |
146 | Let <i>m(G)</i> be the size of a maximum cardinality matching in the graph <i>G</i>. | |
147 | Denote by <i>o(G)</i> the number of connected components in <i>G</i> of odd cardinality, and for a set of | |
148 | vertices <i>X</i>, denote by <i>G - X</i> the induced graph on the vertex set <i>V(G) - X</i>. Then the | |
149 | Tutte-Berge Formula says that | |
150 | <blockquote> | |
151 | <i>2 * m(G) = min ( |V(G)| + |X| - o(G-X) )</i> | |
152 | </blockquote> | |
153 | Where the minimum is taken over all subsets <i>X</i> of the vertex set <i>V(G)</i>. A side effect of the | |
154 | Edmonds Blossom-Shrinking algorithm is that it computes what is known as the Edmonds-Gallai decomposition | |
155 | of a graph: it decomposes the graph into three disjoint sets of vertices, one of which achieves the minimum | |
156 | in the Tutte-Berge Formula. | |
157 | ||
158 | An outline of our verification procedure is: | |
159 | ||
160 | Given a <tt>Graph g</tt> and <tt>MateMap mate</tt>, | |
161 | <ol> | |
162 | <li>Check to make sure that <tt>mate</tt> is a valid matching on <tt>g</tt>. | |
163 | <li>Run <tt>edmonds_augmenting_path_finder</tt> once on <tt>g</tt> and <tt>mate</tt>. If it finds an augmenting | |
164 | path, the matching isn't a maximum cardinality matching. Otherwise, we retrieve a copy of the <tt>vertex_state</tt> | |
165 | map used by the <tt>edmonds_augmenting_path_finder</tt>. The Edmonds-Gallai decomposition tells us that the set | |
166 | of vertices labeled <tt>V_ODD</tt> by the <tt>vertex_state</tt> map can be used as the set X to achieve the | |
167 | minimum in the Tutte-Berge Formula. | |
168 | <li>Count the number of vertices labeled <tt>V_ODD</tt>, store this in <tt>num_odd_vertices</tt>. | |
169 | <li>Create a <a href="filtered_graph.html"><tt>filtered_graph</tt></a> | |
170 | consisting of all vertices that aren't labeled <tt>V_ODD</tt>. Count the number of odd connected components | |
171 | in this graph and store the result in <tt>num_odd_connected_components</tt>. | |
172 | <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. | |
173 | </ol> | |
174 | ||
175 | Assuming these steps are implemented correctly, the verifier will never return a false positive, | |
176 | and will only return a false negative if <tt>edmonds_augmenting_path_finder</tt> doesn't compute the | |
177 | <tt>vertex_state</tt> map correctly, in which case the <tt>edmonds_augmenting_path_finder</tt> | |
178 | isn't working correctly. | |
179 | ||
180 | ||
181 | <h4>Creating Your Own Matching Algorithms</h4> | |
182 | ||
183 | Creating a matching algorithm is as simple as plugging the algorithms described above into a generic | |
184 | matching function, which has the following signature: | |
185 | <pre> | |
186 | template <typename Graph, | |
187 | typename MateMap, | |
188 | typename VertexIndexMap, | |
189 | template <typename, typename, typename> class AugmentingPathFinder, | |
190 | template <typename, typename> class InitialMatchingFinder, | |
191 | template <typename, typename, typename> class MatchingVerifier> | |
192 | bool matching(const Graph& g, MateMap mate, VertexIndexMap vm) | |
193 | </pre> | |
194 | The matching functions provided for you are just inlined specializations of this function: | |
195 | <tt>edmonds_maximum_cardinality_matching</tt> uses <tt>edmonds_augmenting_path_finder</tt> | |
196 | as the <tt>AugmentingPathFinder</tt>, <tt>extra_greedy_matching</tt> as the <tt>InitialMatchingFinder</tt>, | |
197 | and <tt>no_matching_verifier</tt> as the <tt>MatchingVerifier</tt>. | |
198 | <tt>checked_edmonds_maximum_cardinality_matching</tt> uses the same parameters except that | |
199 | <tt>maximum_cardinality_matching_verifier</tt> is used for the <tt>MatchingVerifier</tt>. | |
200 | ||
201 | <p> | |
202 | These aren't necessarily the best choices for any situation - for example, it's been claimed in the literature | |
203 | that for sparse graphs, Edmonds' algorithm converges to the maximum cardinality matching more quickly if it | |
204 | isn't supplied with an intitial matching. Such an algorithm can be easily assembled by calling <tt>matching</tt> with | |
205 | <ul> | |
206 | <li><tt>AugmentingPathFinder = edmonds_augmenting_path_finder</tt> | |
207 | <li><tt>InitialMatchingFinder = empty_matching</tt> | |
208 | </ul> | |
209 | and choosing the <tt>MatchingVerifier</tt> depending on how careful you're feeling. | |
210 | ||
211 | <p> | |
212 | Suppose instead that you want a relatively large matching quickly, but are not exactly interested in a maximum matching. | |
213 | Both extra_greedy_matching and greedy_matching find maximal matchings, which means they're guaranteed to be at | |
214 | least half the size of a maximum cardinality matching, so you could call <tt>matching</tt> with | |
215 | <ul> | |
216 | <li><tt>AugmentingPathFinder = no_augmenting_path_finder</tt> | |
217 | <li><tt>InitialMatchingFinder = extra_greedy_matching</tt> | |
218 | <li><tt>MatchingVerifier = maximum_cardinality_matching_verifier</tt> | |
219 | </ul> | |
220 | The resulting algorithm will find an extra greedy matching in time <i>O(m log n)</i> without looking for | |
221 | augmenting paths. As a bonus, the return value of this function is true if and only if the extra greedy | |
222 | matching happens to be a maximum cardinality matching. | |
223 | ||
224 | </p><h3>Where Defined</h3> | |
225 | ||
226 | <p> | |
227 | <a href="../../../boost/graph/max_cardinality_matching.hpp"><tt>boost/graph/max_cardinality_matching.hpp</tt></a> | |
228 | ||
229 | ||
230 | </p><h3>Parameters</h3> | |
231 | ||
232 | IN: <tt>const Graph& g</tt> | |
233 | <blockquote> | |
234 | An undirected graph. The graph type must be a model of | |
235 | <a href="VertexAndEdgeListGraph.html">Vertex and Edge List Graph</a> and | |
236 | <a href="IncidenceGraph.html">Incidence Graph</a>.<br> | |
237 | </blockquote> | |
238 | ||
239 | IN: <tt>VertexIndexMap vm</tt> | |
240 | <blockquote> | |
241 | Must be a model of <a href="../../property_map/doc/ReadablePropertyMap.html">ReadablePropertyMap</a>, mapping vertices to integer indices. | |
242 | </blockquote> | |
243 | ||
244 | OUT: <tt>MateMap mate</tt> | |
245 | <blockquote> | |
246 | Must be a model of <a href="../../property_map/doc/ReadWritePropertyMap.html">ReadWritePropertyMap</a>, mapping | |
247 | 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 | |
248 | <tt>graph_traits<Graph>::null_vertex()</tt> if v isn't matched. | |
249 | </blockquote> | |
250 | ||
251 | <h3>Complexity</h3> | |
252 | ||
253 | <p> | |
254 | Let <i>m</i> and <i>n</i> be the number of edges and vertices in the input graph, respectively. Assuming the | |
255 | <tt>VertexIndexMap</tt> supplied allows constant-time lookups, the time complexity for both | |
256 | <tt>edmonds_matching</tt> and <tt>checked_edmonds_matching</tt> is <i>O(mn alpha(m,n))</i>. | |
257 | <i>alpha(m,n)</i> is a slow growing function that is at most 4 for any feasible input. | |
258 | </p><p> | |
259 | ||
260 | </p><h3>Example</h3> | |
261 | ||
262 | <p> The file <a href="../example/matching_example.cpp"><tt>example/matching_example.cpp</tt></a> | |
263 | contains an example. | |
264 | ||
265 | <br> | |
266 | </p><hr> | |
267 | <table> | |
268 | <tbody><tr valign="top"> | |
269 |