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Deterministic Padded Decompositions and Negative-Weight Shortest Paths

Jason Li

TL;DR

This work resolves the open problem of a deterministic near-linear-time algorithm for negative-weight single-source shortest paths on directed graphs with integral weights by introducing a padded decomposition that replaces probabilistic low-diameter decompositions. The approach combines a padding-based partitioning with a scaling framework and constructs an auxiliary graph to obtain a valid potential, enabling recovery of shortest-path distances via a Bellman-Ford/Dijkstra hybrid. The main result is a deterministic algorithm running in $O\big((m+n\log\log n)\log(nW)\log^3 n\big)$ time, with independent work offering alternative derandomization via a path cover. This advances deterministic graph algorithms for negative weights and suggests padding-based methods may generalize to broader problems in this domain.

Abstract

We obtain the first near-linear time deterministic algorithm for negative-weight single-source shortest paths on integer-weighted graphs. Our main ingredient is a deterministic construction of a padded decomposition on directed graphs, which may be of independent interest.

Deterministic Padded Decompositions and Negative-Weight Shortest Paths

TL;DR

This work resolves the open problem of a deterministic near-linear-time algorithm for negative-weight single-source shortest paths on directed graphs with integral weights by introducing a padded decomposition that replaces probabilistic low-diameter decompositions. The approach combines a padding-based partitioning with a scaling framework and constructs an auxiliary graph to obtain a valid potential, enabling recovery of shortest-path distances via a Bellman-Ford/Dijkstra hybrid. The main result is a deterministic algorithm running in time, with independent work offering alternative derandomization via a path cover. This advances deterministic graph algorithms for negative weights and suggests padding-based methods may generalize to broader problems in this domain.

Abstract

We obtain the first near-linear time deterministic algorithm for negative-weight single-source shortest paths on integer-weighted graphs. Our main ingredient is a deterministic construction of a padded decomposition on directed graphs, which may be of independent interest.

Paper Structure

This paper contains 10 sections, 17 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Independent work.
  3. Our Techniques
  4. Preliminaries
  5. Negative-Weight Shortest Paths
  6. The algorithm on graph $G^{W/2}$ with recursive parameters $(X,d)$:
  7. Padded Decomposition
  8. Light case
  9. Heavy case
  10. Combining both cases

Key Result

Theorem 1

Given a graph with integral edge weights at least $-W$, there is a deterministic algorithm that computes single-source shortest paths in $O((m+n\log\log n)\log(nW)\log^3n)$ time.

Theorems & Definitions (29)

  • Theorem 1
  • Lemma 2: Hybrid Bellman-Ford/Dijkstra
  • Theorem 3: Padded decomposition
  • Theorem 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 19 more