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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]
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 maximal, but not every maximal matching is a maximum matching.
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.
In any edge-weighted bipartite graph, the maximum w-weight of a matching equals the smallest number of vertices in a w-vertex-cover. The maximum w-weight of a fractional matching is given by the LP: [18] Maximize w · x. Subject to: x ≥ 0 E _____ A G · x ≤ 1 V.
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.
Matching (graph theory) MaxDDBS; Maximal independent set; Maximum agreement subtree problem; Maximum common edge subgraph; Maximum common induced subgraph; Maximum cut; Maximum flow problem; Maximum weight matching; Metric k-center; Minimum k-cut; Mixed Chinese postman problem; Multi-trials technique
The objective is to find an assignment where the sum of qualifications is as big as possible. This problem is identical to finding a maximum-weight matching in an edge-weighted bipartite graph where the nodes of one side arrive online in random order. Thus, it is a special case of the online bipartite matching problem.