Fine-Tuning Decomposition Theorem for Maximum Weight Bipartite Matching

Abstract

Let G be an undirected bipartite graph with non-negative integer weights on the edges. We refine the existing decomposition theorem originally proposed by Kao et al. in the context of maximum weight bipartite matching. We apply it to design an efficient version of the decomposition algorithm to compute the weight of a maximum weight bipartite matching of G in $O(\sqrt{n}W'/k(n,W'/{m'}))$-time by employing an algorithm designed by Feder and Motwani as a subroutine, where n, m, m′( ≤ m) denote number of nodes, number of edges and number of distinct edge weights of G, respectively. The parameter W′ is smaller than the total edge weight W, essentially when the largest edge weight differs by more than one from the second largest edge weight in the current working graph in decomposition step of the algorithm. In best case W′ = O(m) and in worst case W′ = W, i.e., m ≤ W′ ≤ W.