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In an unweighted bipartite graph, the optimization problem is to find a maximum cardinality matching. The problem is solved by the Hopcroft-Karp algorithm in time O ( √ V E ) time, and there are more efficient randomized algorithms , approximation algorithms , and algorithms for special classes of graphs such as bipartite planar graphs , as ...
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 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 the maximum weight matching problem is the assignment problem , in which the graph is a bipartite graph and the matching must have cardinality equal to ...
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. If each vertex can be matched to several vertices at once, then this is a generalized assignment problem. A priority matching is a particular maximum ...
An augmenting path in a matching problem is closely related to the augmenting paths arising in maximum flow problems, paths along which one may increase the amount of flow between the terminals of the flow. It is possible to transform the bipartite matching problem into a maximum flow instance, such that the alternating paths of the matching ...
Longest path problem [3]: ND29 Maximum bipartite subgraph or (especially with weighted edges) maximum cut. [2] [3]: GT25, ND16 Maximum common subgraph isomorphism problem [3]: GT49 Maximum independent set [3]: GT20 Maximum Induced path [3]: GT23 Minimum maximal independent set a.k.a. minimum independent dominating set [4]
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.
In the theory of streaming algorithms for graph matching (for instance to match internet advertisers with advertising slots), the quality of matching covers (sparse subgraphs that approximately preserve the size of a matching in all vertex subsets) is closely related to the density of bipartite graphs that can be partitioned into induced ...