# Matching (graph theory)

The host was quick to respond. Works with different titles in the UK and US. Such a great value for your money! A vertex is matched or saturated if it is an endpoint of one of the edges in the matching. However, there exists a fully polynomial time randomized approximation scheme for counting the number of bipartite matchings.

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Matching problems are often concerned with bipartite graphs. An alternative randomized approach is based on the fast matrix multiplication algorithm and gives O V 2.

In a weighted bipartite graph, each edge has an associated value. A maximum weighted bipartite matching [4] is defined as a matching where the sum of the values of the edges in the matching have a maximal value. If the graph is not complete bipartite , missing edges are inserted with value zero. Finding such a matching is known as 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. A generalization of the same technique can also be used to find maximum independent sets in claw-free graphs.

Another randomized algorithm by Mucha and Sankowski, [5] based on the fast matrix multiplication algorithm, gives O V 2. A maximal matching can be found with a simple greedy algorithm. A maximum matching is also a maximal matching, and hence it is possible to find a largest maximal matching in polynomial time. However, no polynomial-time algorithm is known for finding a minimum maximal matching , that is, a maximal matching that contains the smallest possible number of edges. Note that 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. 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. However, there exists a fully polynomial time randomized approximation scheme for counting the number of bipartite matchings. 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. The problem of developing an online algorithm for matching was first considered by Karp et al.

This is a natural generalization of the secretary problem and has applications to online ad auctions. Via this result, the minimum vertex cover, maximum independent set , and maximum vertex biclique problems may be solved in polynomial time for bipartite graphs. Hall's marriage theorem provides a characterization of bipartite graphs which have a perfect matching and the Tutte theorem provides a characterization for arbitrary graphs.

A perfect matching is a spanning 1-regular subgraph, a. In general, a spanning k -regular subgraph is a k -factor. From Wikipedia, the free encyclopedia. Redirected from Maximum matching. For comparisons of two graphs, see Graph matching. Foundations of Computer Science , pp. Minimum edge dominating set optimisation version is the problem GT3 in Appendix B page Minimum maximal matching optimisation version is the problem GT10 in Appendix B page Check date values in: Retrieved from " https: