All-cavity maximum matchings

Abstract

Let G = (X, Y, E) be a bipartite graph with integer weights on the edges. Let n, m, and N denote the vertex count, the edge count, and an upper bound on the absolute values of edge weights of G, respectively. For a vertex u in G, let Gu denote the graph formed by deleting u from G. The all-cavity maximum matching problem asks for a maximum weight matching in Gu for all u in G. This problem finds applications in optimal tree algorithms for computational biology. We show that the problem is solvable in. (formula presented) time, matching the currently best time complexity for merely computing a single maximum weight matching in G. We also give an algorithm for a generalization of the problem where both a vertex from X and one from Y can be deleted. The running time is (formula presented). Our algorithms are based on novel linear-time reductions among problems of computing shortest paths and all-cavity maximum matchings.