Negative-Weight shortest paths and unit capacity minimum cost flow in Õ(m[superscript 10/7] log W) Time

Abstract

In this paper, we study a set of combinatorial optimization problems on weighted graphs: the shortest path problem with negative weights, the weighted perfect bipartite matching problem, the unit-capacity minimum-cost maximum flow problem, and the weighted perfect bipartite b-matching problem under the assumption that ||b||1 = O(m). We show that each of these four problems can be solved in Õ(m[superscript 10/7] log W) time, where W is the absolute maximum weight of an edge in the graph, providing the first polynomial improvement in their sparse-graph time complexity in over 25 years. At a high level, our algorithms build on the interior-point method-based framework developed by Mądry (FOCS 2013) for solving unit-capacity maximum flow problem. We develop a refined way to analyze this framework, as well as provide new variants of the underlying preconditioning and perturbation techniques. Consequently, we are able to extend the whole interior-point method-based approach to make it applicable in the weighted graph regime.