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Bellman-Ford in Almost-Linear Time for Dense Graphs

George Z. Li, Jason Li, Junkai Zhang

TL;DR

The paper tackles the single-source shortest path problem on directed graphs with real-valued, potentially negative, edge weights. It builds on the shortcutting framework to reduce the number of negative edges along shortest paths by adding carefully constructed shortcut edges and using a potent betweenness reduction, now with a stronger SBW notion and multi-scale bookkeeping. The authors present two shortcutting constructions: a simpler n^{7/3+o(1)} method and a refined n^{2+o(1)} approach based on a multi-scale bucketing of negative vertices, Steiner-style augmentations, and copies of vertices, all while preserving distances via potential reweightings. Together, these yield a randomized algorithm for SSSP that runs in n^{2+o(1)} time on dense graphs, marking a substantial improvement over prior techniques for real-weighted negative-edge scenarios. The work contributes both a practical path toward near-linear dense-case runtimes and a stronger theoretical toolset (strongbetweennessSBW) that could influence future refinements in the sparse regime as well.

Abstract

We consider the single-source shortest paths problem on a directed graph with real-valued (possibly negative) edge weights and solve this problem in $n^{2+o(1)}$ time by refining the shortcutting procedure introduced in Li, Li, Rao, and Zhang (2026).

Bellman-Ford in Almost-Linear Time for Dense Graphs

TL;DR

The paper tackles the single-source shortest path problem on directed graphs with real-valued, potentially negative, edge weights. It builds on the shortcutting framework to reduce the number of negative edges along shortest paths by adding carefully constructed shortcut edges and using a potent betweenness reduction, now with a stronger SBW notion and multi-scale bookkeeping. The authors present two shortcutting constructions: a simpler n^{7/3+o(1)} method and a refined n^{2+o(1)} approach based on a multi-scale bucketing of negative vertices, Steiner-style augmentations, and copies of vertices, all while preserving distances via potential reweightings. Together, these yield a randomized algorithm for SSSP that runs in n^{2+o(1)} time on dense graphs, marking a substantial improvement over prior techniques for real-weighted negative-edge scenarios. The work contributes both a practical path toward near-linear dense-case runtimes and a stronger theoretical toolset (strongbetweennessSBW) that could influence future refinements in the sparse regime as well.

Abstract

We consider the single-source shortest paths problem on a directed graph with real-valued (possibly negative) edge weights and solve this problem in time by refining the shortcutting procedure introduced in Li, Li, Rao, and Zhang (2026).
Paper Structure (15 sections, 13 theorems, 28 equations, 3 figures)

This paper contains 15 sections, 13 theorems, 28 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Copies of Vertices
  4. Preprocessing: Unique Incident Negative Edges
  5. Technical Overview
  6. Shortcutting Algorithm
  7. Stronger Betweenness Reduction
  8. Forward and Backward Searches
  9. The Shortcutting Algorithm
  10. A Simpler Construction
  11. Step 1.
  12. Step 2.
  13. Step 3.
  14. The Algorithm
  15. Completing the Shortcutting

Key Result

Theorem 1

There is a randomized algorithm for single-source shortest paths on real-weighted directed graphs which runs in $n^{2+o(1)}$ time.

Figures (3)

  • Figure 1: The three cases in the proof of \ref{['lem:shortcut']}, where height indicates cumulative distance to each vertex. Negative edges are marked in red, and the new shortcut path is marked in bold.
  • Figure 2: An illustration of steps 2 and 3 from \ref{['lem:shortcut-good-paths']}, where height indicates cumulative distance, and points on the same vertical line are copies of the same vertex. The left part shows the case where the edge $(u,v_{in})$ exists for a locally-negative path $u\to v_i\to\tilde{r}$, and such paths are shortcut in step 2. The right part shows the case where this edge does not exist. In this case, $u\in \tilde{V}_{in}(r_{in}(v))$, and such paths are shortcut in step 3.
  • Figure :

Theorems & Definitions (33)

  • Theorem 1
  • Claim 2
  • proof
  • Lemma 3
  • proof
  • Definition 4
  • Lemma 5: Betweenness reduction
  • Lemma 6: Lemma 8 of li2025shortcutting
  • Lemma 7: Lemma 9 of li2025shortcutting
  • Lemma 8
  • ...and 23 more