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Shortcutting for Negative-Weight Shortest Path

George Z. Li, Jason Li, Satish Rao, Junkai Zhang

TL;DR

This work proposes an shortcutting procedure that iteratively reduces the number of negative-weight edges along shortest paths by a constant factor, improving on prior work of Fineman and Huang-Jin-Quanrud on dense graphs.

Abstract

Consider the single-source shortest paths problem on a directed graph with real-valued edge weights. We solve this problem in $O(n^{2.5}\log^{4.5}n)$ time, improving on prior work of Fineman (STOC 2024) and Huang-Jin-Quanrud (SODA 2025, 2026) on dense graphs. Our main technique is an shortcutting procedure that iteratively reduces the number of negative-weight edges along shortest paths by a constant factor.

Shortcutting for Negative-Weight Shortest Path

TL;DR

This work proposes an shortcutting procedure that iteratively reduces the number of negative-weight edges along shortest paths by a constant factor, improving on prior work of Fineman and Huang-Jin-Quanrud on dense graphs.

Abstract

Consider the single-source shortest paths problem on a directed graph with real-valued edge weights. We solve this problem in time, improving on prior work of Fineman (STOC 2024) and Huang-Jin-Quanrud (SODA 2025, 2026) on dense graphs. Our main technique is an shortcutting procedure that iteratively reduces the number of negative-weight edges along shortest paths by a constant factor.

Paper Structure

This paper contains 25 sections, 29 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Techniques
  3. Preliminaries
  4. Recursive Betweenness Reduction
  5. Iterative Hop Reduction
  6. Solving the Recurrence
  7. Reducing Betweenness Using Shortest Path
  8. Improving the betweenness reduction in
  9. Hop reducers and remote sets.
  10. Betweenness reduction and negative sandwiches.
  11. $\mathbf{\tilde{O}(mn^{4/5})}$ time algorithm.
  12. Proof of \ref{['thm:improved_sssp']}.
  13. Refining
  14. Betweenness Reduction.
  15. Negative Sandwiches.
  16. ...and 10 more sections

Key Result

Theorem 1

There is a randomized algorithm for single-source shortest paths on real-weighted graphs that runs in $O(n^{2.5}\log^{4.5}n)$ time.

Figures (1)

  • Figure 1: The three cases of shortcutting in the proof of \ref{['lem:shortcut']}, where the height indicates cumulative distance to each vertex. Negative edges are marked in red, and the new shortcut path is marked in bold.

Theorems & Definitions (46)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • proof
  • Lemma 5: Betweenness reduction
  • proof
  • Theorem 6
  • Corollary 7
  • Lemma 8
  • ...and 36 more