An $n^{2+o(1)}$ Time Algorithm for Single-Source Negative Weight Shortest Paths

TL;DR

This paper introduces a randomized algorithm that computes single-source shortest paths in directed graphs with real weights in near-quadratic time, significantly improving efficiency for dense graphs and leveraging novel graph compression techniques.

Contribution

The paper presents the first almost linear-time algorithm for dense graphs in the SSSP problem, utilizing a new compression method with auxiliary Steiner vertices to efficiently generate shortcuts.

Findings

Achieves $n^{2+o(1)}$ time complexity for SSSP.

First nearly linear-time algorithm for dense graphs.

Introduces a novel graph compression technique with Steiner vertices.

Abstract

We present a randomized algorithm for the single-source shortest paths (SSSP) problem on directed graphs with arbitrary real-valued edge weights that runs in $n^{2+o(1)}$ time with high probability. This result yields the first almost linear-time algorithm for the problem on dense graphs ($m = \Theta(n^2)$) and improves upon the best previously known bounds for moderately dense graphs ($m = \omega(n^{1.306})$). Our approach builds on the hop-reduction via shortcutting framework introduced by Li, Li, Rao, and Zhang (2025), which iteratively augments the graph with shortcut edges to reduce the negative hop count of shortest paths. The central computational bottleneck in prior work is the cost of explicitly constructing these shortcuts in dense regions. We overcome this by introducing a new compression technique using auxiliary Steiner vertices. Specifically, we construct these vertices to represent large neighborhoods compactly in a structured manner, allowing us to efficiently generate and propagate shortcuts while strictly controlling the growth of vertex degrees and graph size.