A Decomposition Theorem for Maximum Weight Bipartite Matchings

TL;DR

This paper introduces a new decomposition theorem for maximum weight bipartite matchings and presents an efficient algorithm that significantly improves the computational complexity for solving this problem.

Contribution

The paper proposes a novel decomposition theorem and an improved algorithm that bridges the gap between weight and cardinality matching complexities in bipartite graphs.

Findings

New decomposition theorem for maximum weight bipartite matchings

An O(sqrt(n)W/k(n,W/N)) time algorithm for maximum weight matching

Efficient computation of maximum weight matchings after node removal

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.