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A Decomposition Theorem for Maximum Weight Bipartite Matchings

Ming-Yang Kao·Tak-Wah Lam·Wing-Kin Sung·Hing-Fung Ting
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Abstract

Let G be a bipartite graph with positive integer weights on the edges and without isolated nodes. Let n, N and W be the node count, the largest edge weight and the total weight of G. Let k(x,y) be log(x)/log(x^2/y). We present a new decomposition theorem for maximum weight bipartite matchings and use it to design an O(sqrt(n)W/k(n,W/N))-time algorithm for computing a maximum weight matching of G. This algorithm bridges a long-standing gap between the best known time complexity of computing a maximum weight matching and that of computing a maximum cardinality matching. Given G and a maximum weight matching of G, we can further compute the weight of a maximum weight matching of G-{u} for all nodes u in O(W) time.

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Ming-Yang Kao, Tak-Wah Lam, Wing-Kin Sung, Hing-Fung Ting. 2000-11-11. A Decomposition Theorem for Maximum Weight Bipartite Matchings. https://arxiv.org/abs/cs/0011015

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