A Decomposition Theorem for Maximum Weight Bipartite Matchings

  • Authors:

  • Ming-Yang Kao;Tak-Wah Lam;Wing-Kin Sung;Hing-Fung Ting

  • Affiliations:

  • -;-;-;-

  • Venue:
  • SIAM Journal on Computing
  • Year:
  • 2002

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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 (x2/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.