All-cavity maximum matchings

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 O(√nmlog(nN)) 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 O(n21og n + nm). Our algorithms are based on novel linear-time reductions among problems of computing shortest paths and all-cavity maximum matchings.