Matching (graph Theory)
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Matching Graph Theory









A perfect matching (a.k.a. 1-factor) is a matching which matches all vertices of the graph. That is, every vertex of the graph is incident to exactly one edge of the matching. Every perfect matching is maximum and hence maximal. In some literature, the term complete matching is used. In the above figure, only part (b) shows a perfect matching. A perfect matching is also a minimum-size edge cover. Thus, ν(G) ≤ ρ(G) , that is, the size of a maximum matching is no larger than the size of a minimum edge cover. A perfect matching can only occur when the graph has an even number of vertices.


Given a matching M,



An induced matching is a matching that is an induced subgraph.[3]


In any graph without isolated vertices, the sum of the matching number and the edge covering number equals the number of vertices.[4] If there is a perfect matching, then both the matching number and the edge cover number are |V | / 2.

If A and B are two maximal matchings, then |A| B| and |B| A|. To see this, observe that each edge in B \ A can be adjacent to at most two edges in A \ B because A is a matching; moreover each edge in A \ B is adjacent to an edge in B \ A by maximality of B, hence

Further we deduce that


Matching polynomials


Another definition gives the matching polynomial as


Algorithms and computational complexity

Maximum-cardinality matching

A fundamental problem in combinatorial optimization is finding a maximum matching. This problem has various algorithms for different classes of graphs.

In an unweighted bipartite graph, the optimization problem is to find a maximum cardinality matching. The problem is solved by the Hopcroft-Karp algorithm in time O(E) time. A randomised algorithm by Mucha and Sankowski, based on the fast matrix multiplication algorithm, gives complexity. For the special case of planar graphs the problem can be solved in time .

Maximum-weight matching

In a weighted bipartite graph, the optimization problem is to find a maximum-weight matching; a dual problem is to find a minimum-weight matching. This problem is often called maximum weighted bipartite matching, or the assignment problem. The Hungarian algorithm solves the assignment problem and it was one of the beginnings of combinatorial optimization algorithms. It uses a modified shortest path search in the augmenting path algorithm. If the Bellman-Ford algorithm is used for this step, the running time of the Hungarian algorithm becomes , or the edge cost can be shifted with a potential to achieve running time with the Dijkstra algorithm and Fibonacci heap.[5]

In a non-bipartite weighted graph, the problem of maximum weight matching can be solved in time using Edmonds' blossom algorithm.

Maximal matchings


A maximal matching with k edges is an edge dominating set with k edges. Conversely, if we are given a minimum edge dominating set with k edges, we can construct a maximal matching with k edges in polynomial time. Therefore, the problem of finding a minimum maximal matching is essentially equal to the problem of finding a minimum edge dominating set.[6] Both of these two optimization problems are known to be NP-hard; the decision versions of these problems are classical examples of NP-complete problems.[7] Both problems can be approximated within factor 2 in polynomial time: simply find an arbitrary maximal matching M.[8]

Counting problems

The number of matchings in a graph is known as the Hosoya index of the graph. It is #P-complete to compute this quantity, even for bipartite graphs.[9] It is also #P-complete to count perfect matchings, even in bipartite graphs, because computing the permanent of an arbitrary 0-1 matrix (another #P-complete problem) is the same as computing the number of perfect matchings in the bipartite graph having the given matrix as its biadjacency matrix. However, there exists a fully polynomial time randomized approximation scheme for counting the number of bipartite matchings.[10] A remarkable theorem of Kasteleyn states that the number of perfect matchings in a planar graph can be computed exactly in polynomial time via the FKT algorithm.

The number of perfect matchings in a complete graph Kn (with n even) is given by the double factorial (n − 1)!!.[11] The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are given by the telephone numbers.[12]

Finding all maximally-matchable edges

One of the basic problems in matching theory is to find in a given graph all edges that may be extended to a maximum matching in the graph (such edges are called maximally-matchable edges, or allowed edges). Algorithms for this problem include:

  • For general graphs, a deterministic algorithm in time and a randomized algorithm in time .
  • For bipartite graphs, if a single maximum matching is found, a deterministic algorithm runs in time .

Online bipartite matching

The problem of developing an online algorithm for matching was first considered by Karp et al.[13] In the online setting, nodes on one side of the bipartite graph arrive one at a time and must either be immediately matched to the other side of the graph or discarded. This is a natural generalization of the secretary problem and has applications to online ad auctions.[14] The best online algorithm, for the unweighted maximization case with a random arrival model, attains a competitive ratio of .[15]

Characterizations and notes





Matching in general graphs

Matching in bipartite graphs

See also

  • Stable marriage problem (also known as stable matching problem)


  2. ^ http://diestel-graph-theory.com/basic.html
  3. ^ Cameron, Kathie (1989), "Induced matchings", Special issue for First Montreal Conference on Combinatorics and Computer Science, 1987, Discrete Applied Mathematics, 24 (1-3): 97-102, doi:10.1016/0166-218X(92)90275-F, MR 1011265
  5. ^ Fredman, Michael L.; Tarjan, Robert Endre (1987), "Fibonacci heaps and their uses in improved network optimization algorithms", Journal of the ACM, 34 (3): 596-615, doi:10.1145/28869.28874
  6. ^ Yannakakis, Mihalis; Gavril, Fanica (1980), "Edge dominating sets in graphs" (PDF), SIAM Journal on Applied Mathematics, 38 (3): 364-372, doi:10.1137/0138030.
  10. ^ Bezáková, Ivona; ?tefankovi?, Daniel; Vazirani, Vijay V.; Vigoda, Eric (2008). "Accelerating Simulated Annealing for the Permanent and Combinatorial Counting Problems". SIAM Journal on Computing. 37 (5): 1429-1454. CiteSeerX doi:10.1137/050644033.
  12. ^ Tichy, Robert F.; Wagner, Stephan (2005), "Extremal problems for topological indices in combinatorial chemistry" (PDF), Journal of Computational Biology, 12 (7): 1004-1013, doi:10.1089/cmb.2005.12.1004.
  13. ^ "An Optimal Algorithm for On-line Bipartite Matching" (PDF). Cite journal requires |journal= (help)
  14. ^ Online Matching and Ad Allocation (PDF). Retrieved 285-299. Check date values in: |accessdate= (help)
  16. ^ See, e.g., Trinajsti?, Nenad; Klein, Douglas J.; Randi?, Milan (1986), "On some solved and unsolved problems of chemical graph theory", International Journal of Quantum Chemistry, 30 (S20): 699-742, doi:10.1002/qua.560300762.

Further reading

  2. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein (2001), Introduction to Algorithms (second ed.), MIT Press and McGraw-Hill, Chapter 26, pp. 643–700, ISBN 0-262-53196-8CS1 maint: multiple names: authors list (link)
  3. András Frank (2004). On Kuhn's Hungarian Method - A tribute from Hungary (PDF) (Technical report). Egerváry Research Group.
  4. Michael L. Fredman and Robert E. Tarjan (1987), "Fibonacci heaps and their uses in improved network optimization algorithms", Journal of the ACM, 34 (3): 595-615, doi:10.1145/28869.28874.

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