Las Vegas RNC algorithms for unary weighted perfect matching and T-join problems

Abstract

Karp, Upfal and Wigderson developed an RNC algorithm to find a minimum-weight perfect matching in a graph when the edge weights are given in unary. This was a Monte Carlo randomized algorithm—it produced an optimal solution with high probability, but could not certify the optimality of its solution. In this paper we give a Las Vegas RNC algorithm for the problem—one that certifies, with high probability, that its solution is correct—utilizing reductions between the minimum- weight perfect matching problem and the T-join problem. We also give Las Vegas RNC algorithms for finding a minimum cardinality T-join, a maximum 1-packing of T-cuts in a bipartite graph, the T-join structure of a graph, and a planar multicommodity flow.