Faster Negative-Weight Shortest Paths and Directed Low-Diameter Decompositions

TL;DR

This paper introduces a faster algorithm for low-diameter decompositions in directed graphs and applies it to improve the efficiency of negative-weight shortest path computations, achieving near log-factor speedups.

Contribution

The paper presents a novel, faster low-diameter decomposition algorithm for directed graphs and leverages it to enhance negative-weight shortest path algorithms.

Findings

Achieves $O((m+n ext{log} ext{log} n) ext{log} n ext{log} ext{log} n)$ expected time for low-diameter decompositions.

Improves negative-weight shortest path algorithm to $O((m+n ext{log} ext{log} n) ext{log}(nW) ext{log} n ext{log} ext{log} n)$ time.

Matches the $O( ext{log} n ext{log} ext{log} n)$ loss factor from prior work.

Abstract

We present a faster algorithm for low-diameter decompositions on directed graphs, matching the $O(\log n\log\log n)$ loss factor from Bringmann, Fischer, Haeupler, and Latypov (ICALP 2025) and improving the running time to $O((m+n\log\log n)\log n\log\log n)$ in expectation. We then apply our faster low-diameter decomposition to obtain an algorithm for negative-weight single source shortest paths on integer-weighted graphs in $O((m+n\log\log n)\log(nW)\log n\log\log n)$ time, a nearly log-factor improvement over the algorithm of Bringmann, Cassis, and Fischer (FOCS 2023).