Negative-Weight Shortest Paths and Unit Capacity Minimum Cost Flow in $\tilde{O}(m^{10/7} \log W)$ Time

TL;DR

This paper presents a significant polynomial time complexity improvement for solving several combinatorial optimization problems on weighted graphs, including shortest paths with negative weights and minimum-cost flow, using an interior-point method framework.

Contribution

It extends the interior-point method framework to weighted graphs and introduces new analysis, preconditioning, and perturbation techniques for faster algorithms.

Findings

All four problems solved in O(m^{10/7}\u221a W) time.

First polynomial improvement in over 25 years for these problems.

Applicable to sparse graphs with bounded  b_1 norm.

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 $\Vert b\Vert_1=O(m)$. We show that each one of these four problems can be solved in $\tilde{O}(m^{10/7}\log W)$ time, where $W$ is the absolute maximum weight of an edge in the graph, which gives the first in over 25 years polynomial improvement in their sparse-graph time complexity. At a high level, our algorithms build on the interior-point method-based framework developed by Madry (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.