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

TL;DR

This work advances directed graph algorithms for negative weights by developing a faster directed low-diameter decomposition (LDD) with loss ℓ(n)=O(log n log log n) and expected time O((m+n log log n) log n log log n). It then employs this LDD within a Scale-based framework to achieve a near-optimal Negative-Weight SSSP algorithm, running in O((m+n log log n) log(nW) log n log log n) time and either producing shortest-path distances or a negative-weight cycle. Core techniques include CKR-style uniform-radius ball sampling, heavy-vertex elimination, and an intricate weaving of LDD recursions with scaling recurrences to bound edge cuts and recursion depth. The results provide near-linear-time performance for directed graphs with negative edges and offer a practical pathway to faster shortest-path computations in scenarios where edge weights can be negative. Overall, the paper tightens the gap to linear-time behavior for directed negative-weight shortest paths and strengthens the toolkit for graph decompositions in negative-weight settings.

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).

Theorems & Definitions (56)

• Theorem 2.1
• Theorem 2.2: Preprocessing
• Remark 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• Lemma 2.7
• proof
• Lemma 2.8