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In computer science and graph theory, the maximum weight matching problem is the problem of finding, in a weighted graph, a matching in which the sum of weights is maximized. A special case of it is the assignment problem , in which the input is restricted to be a bipartite graph , and the matching constrained to be have cardinality that of the ...
The matching problem can be generalized by assigning weights to edges in G and asking for a set M that produces a matching of maximum (minimum) total weight: this is the maximum weight matching problem. This problem can be solved by a combinatorial algorithm that uses the unweighted Edmonds's algorithm as a subroutine. [6]
When phrased as a graph theory problem, the assignment problem can be extended from bipartite graphs to arbitrary graphs. The corresponding problem, of finding a matching in a weighted graph where the sum of weights is maximized, is called the maximum weight matching problem.
The following figure shows examples of maximal matchings (red) in three graphs. A maximum matching (also known as maximum-cardinality matching [2]) is a matching that contains the largest possible number of edges. There may be many maximum matchings. The matching number of a graph G is the size of a maximum matching. Every maximum matching is ...
An example of a bipartite graph, with a maximum matching (blue) and minimum vertex cover (red) both of size six. In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig (), describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs.
The original form of the auction algorithm is an iterative method to find the optimal prices and an assignment that maximizes the net benefit in a bipartite graph, the maximum weight matching problem (MWM). [2] [3] This algorithm was first proposed by Dimitri Bertsekas in 1979.
The fifth corner (1/2,1/2,1/2) does not represent a matching - it represents a fractional matching in which each edge is "half in, half out". Note that this is the largest fractional matching in this graph - its weight is 3/2, in contrast to the three integral matchings whose size is only 1. As another example, in the 4-cycle there are 4 edges.
An edge dominating set for G is a dominating set for its line graph L(G) and vice versa. Any maximal matching is always an edge dominating set. Figures (b) and (d) are examples of maximal matchings. Furthermore, the size of a minimum edge dominating set equals the size of a minimum maximal matching. A minimum maximal matching is a minimum edge ...