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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.
The assignment problem consists of finding, in a weighted bipartite graph, a matching of maximum size, in which the sum of weights of the edges is minimum. If the numbers of agents and tasks are equal, then the problem is called balanced assignment , and the graph-theoretic version is called minimum-cost perfect matching .
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 problem of finding a maximum vertex-packing in a graph is equivalent to the problem of finding a maximum matching in a hypergraph: [1]: 467 Given a hypergraph H = (V, E), define its intersection graph Int(H) as the simple graph whose vertices are E and whose edges are pairs (e 1,e 2) such that e 1, e 2 have a vertex in common
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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 ...