 
Summary: Fast algorithms for Maximum Subset Matching and AllPairs
Shortest Paths in graphs with a (not so) small vertex cover
Noga Alon
Raphael Yuster
Abstract
In the Maximum Subset Matching problem, which generalizes the maximum matching prob
lem, 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 ~O(ms(1)/2
) time if m s(+1)/2
and in
~O(s
) time if m s(+1)/2
, where < 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 AllPairs 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 ~O(Wn(1,1,µ)
