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Weighted Matchings via Unweighted Augmentations

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Ola Nils Anders Svensson, Slobodan Mitrovic, Buddhima Ruwanmini Gamlath Gamlath Ralalage
2019

Abstract

We design a generic method to reduce the task of finding weighted matchings to that of finding short augmenting paths in unweighted graphs. This method enables us to provide efficient implementations for approximating weighted matchings in the massively parallel computation (MPC) model and in the streaming model.

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https://infoscience.epfl.ch/entities/publication/2d69cefe-e0a7-47cb-b802-bbcb4f9470b2

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Matching (graph theory)

In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a network flow problem. Given a graph G = (V, E), a matching M in G is a set of pairwise non-adjacent edges, none of which are loops; that is, no two edges share common vertices.

Maximum weight 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 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 smaller of the two partitions. Another special case is the problem of finding a maximum cardinality matching on an unweighted graph: this corresponds to the case where all edge weights are the same.

Kőnig's theorem (graph theory)

In the mathematical area of graph theory, Kőnig's theorem, proved by , describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs. It was discovered independently, also in 1931, by Jenő Egerváry in the more general case of weighted graphs. A vertex cover in a graph is a set of vertices that includes at least one endpoint of every edge, and a vertex cover is minimum if no other vertex cover has fewer vertices.

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