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

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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.