Search results
Results from the WOW.Com Content Network
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 ...
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.
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.
In the unbalanced assignment problem, the larger part of the bipartite graph has n vertices and the smaller part has r<n vertices. There is also a constant s which is at most the cardinality of a maximum matching in the graph. The goal is to find a minimum-cost matching of size exactly s.
By finding a maximum-cardinality matching, it is possible to decide whether there exists a perfect matching. The problem of finding a matching with maximum weight in a weighted graph is called the maximum weight matching problem, and its restriction to bipartite graphs is called the assignment problem.
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]
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.
Every perfect matching is a maximum-cardinality matching, but the opposite is not true. For example, consider the following graphs: [1] In graph (b) there is a perfect matching (of size 3) since all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are ...