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Negative-Weight Single-Source Shortest Paths in Near-linear Time

Aaron Bernstein, Danupon Nanongkai, Christian Wulff-Nilsen

TL;DR

The paper resolves the classic problem of single-source shortest paths in directed graphs with integral, potentially negative weights by presenting a simple, combinatorial Las Vegas algorithm with near-linear running time: $O(m\log^8(n)\log W)$. Central to the approach are a recursive scaling framework (ScaleDown) and a Low-Diameter Decomposition (LDD) that partitions the graph into small-diameter SCCs, together with price-function transformations and a black-box reduction to handle negative cycles. The combination enables efficient non-negativity adjustments and a final ElimNeg phase to guarantee non-negative edge weights, allowing a final Dijkstra run to produce a shortest-path tree when no negative cycle exists. This yields near-linear performance without relying on advanced continuous optimization or dynamic methods, marking a significant simplification and strengthening of negative-weight SSSP techniques and offering a path toward broader applications in fundamental graph problems.

Abstract

We present a randomized algorithm that computes single-source shortest paths (SSSP) in $O(m\log^8(n)\log W)$ time when edge weights are integral and can be negative. This essentially resolves the classic negative-weight SSSP problem. The previous bounds are $\tilde O((m+n^{1.5})\log W)$ [BLNPSSSW FOCS'20] and $m^{4/3+o(1)}\log W$ [AMV FOCS'20]. Near-linear time algorithms were known previously only for the special case of planar directed graphs [Fakcharoenphol and Rao FOCS'01]. In contrast to all recent developments that rely on sophisticated continuous optimization methods and dynamic algorithms, our algorithm is simple: it requires only a simple graph decomposition and elementary combinatorial tools. In fact, ours is the first combinatorial algorithm for negative-weight SSSP to break through the classic $\tilde O(m\sqrt{n}\log W)$ bound from over three decades ago [Gabow and Tarjan SICOMP'89].

Negative-Weight Single-Source Shortest Paths in Near-linear Time

TL;DR

The paper resolves the classic problem of single-source shortest paths in directed graphs with integral, potentially negative weights by presenting a simple, combinatorial Las Vegas algorithm with near-linear running time: . Central to the approach are a recursive scaling framework (ScaleDown) and a Low-Diameter Decomposition (LDD) that partitions the graph into small-diameter SCCs, together with price-function transformations and a black-box reduction to handle negative cycles. The combination enables efficient non-negativity adjustments and a final ElimNeg phase to guarantee non-negative edge weights, allowing a final Dijkstra run to produce a shortest-path tree when no negative cycle exists. This yields near-linear performance without relying on advanced continuous optimization or dynamic methods, marking a significant simplification and strengthening of negative-weight SSSP techniques and offering a path toward broader applications in fundamental graph problems.

Abstract

We present a randomized algorithm that computes single-source shortest paths (SSSP) in time when edge weights are integral and can be negative. This essentially resolves the classic negative-weight SSSP problem. The previous bounds are [BLNPSSSW FOCS'20] and [AMV FOCS'20]. Near-linear time algorithms were known previously only for the special case of planar directed graphs [Fakcharoenphol and Rao FOCS'01]. In contrast to all recent developments that rely on sophisticated continuous optimization methods and dynamic algorithms, our algorithm is simple: it requires only a simple graph decomposition and elementary combinatorial tools. In fact, ours is the first combinatorial algorithm for negative-weight SSSP to break through the classic bound from over three decades ago [Gabow and Tarjan SICOMP'89].
Paper Structure (51 sections, 35 theorems, 35 equations)

This paper contains 51 sections, 35 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Our Result
  3. Perspective on Other Problems:
  4. Independent and Follow-Up Results.
  5. Techniques
  6. Low-Diameter Decomposition.
  7. No Negative-Weight Cycle Assumption via a Black-Box Reduction.
  8. Preliminaries
  9. Dummy Source and Negative Edges.
  10. Price Functions and Equivalence
  11. The Framework
  12. Basic Subroutines
  13. The Interface of the Two Main Algorithms
  14. Remark: Termination and Negative-Weight Cycles.
  15. Proof sketch of Theorem \ref{['thm:result']}.
  16. ...and 36 more sections

Key Result

theorem 1.1

There exists a randomized (Las Vegas) algorithm that takes $O(m\log^8(n)\log(W))$ time with high probability (and in expectation) for an $m$-edge input graph $G_{in}$ and source $s_{in}$. It either returns a shortest path tree from $s_{in}$ or returns a negative-weight cycle.

Theorems & Definitions (82)

  • theorem 1.1
  • lemma 1.1
  • lemma 2.2
  • proof
  • definition 2.3: $G_s$,$w_s$,$G^B$,$w^B$,$G^B_s$,$w^B_s$
  • definition 2.4: $\eta_G(v)$, $P_G(v)$
  • definition 2.5: Price Function
  • definition 2.6: Graph Equivalence
  • lemma 2.7: Johnson77
  • lemma 2.8: Johnson77
  • ...and 72 more