Fast algorithms for maximum subset matching and all-pairs shortest paths in graphs with a (not so) small vertex cover

Abstract

In the Maximum Subset Matching problem, which generalizes the maximum matching problem, we are given a graph G = (V, E) and S ⊂ V. The goal is to determine the maximum number of vertices of S that can be matched in a matching of G. Our first result is a new randomized algorithm for the Maximum Subset Matching problem that improves upon the fastest known algorithms for this problem. Our algorithm runs in Ṏ(ms(ω-1)/2) time if m > s(ω+1)/2 and in Õ(sω) time if m < 2.376 is the matrix multiplication exponent, m is the number of edges from S to V \ S, and s = |S|. The algorithm is based, in part, on a method for computing the rank of sparse rectangular integer matrices. Our second result is a new algorithm for the All-Pairs Shortest Paths (APSP) problem. Given an undirected graph with n vertices, and with integer weights from (1,...,W) assigned to its edges, we present an algorithm that solves the APSP problem in Õ(Wn ω(1,1,μ)) time where nμ = vc(G) is the vertex cover number of G and ω(1, 1, μ) is the time needed to compute the Boolean product of an n × n matrix with an n × nμ matrix. Already for the unweighted case this improves upon the previous O(n 2+μ) and Õ(nω) time algorithms for this problem. In particular, if a graph has a vertex cover of size O(n 0.29) then APSP in unweighted graphs can be solved in asymptotically optimal Õ(n2) time, and otherwise it can be solved in O(n 1.844vc(G)0.533) time. The common feature of both results is their use of algorithms developed in recent years for fast (sparse) rectangular matrix multiplication. © Springer-Verlag Berlin Heidelberg 2007.