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

TL;DR

This work 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 and introduces a new compression technique using auxiliary 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 ($m = Θ(n^2)$) and improves upon the best previously known bounds for moderately dense graphs ($m = ω(n^{1.306})$). 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.

Figures (3)

• Figure 1: Illustration of two simple cases. The height of each vertex represents its cumulative distance along the path. Black edges denote non-negative edges in the graph, red edges denote negative edges, and purple edges are the shortcut edges added by the shortcut algorithm.
• Figure 2: Illustration of the hard case, and an in-Steiner gadget.
• Figure 3: Illustration of some steps in the procedure $\mathtt{ShortcutIn}(r,t)$.

Theorems & Definitions (59)

• Theorem 1.1
• Lemma 3.1: Lemma 11 of LLRZ25
• Lemma 3.2: Lemma 8 of LLRZ25
• Lemma 3.3
• Claim 3.4
• proof
• Claim 3.5
• proof
• Remark 4.1
• proof