Deterministic Negative-Weight Shortest Paths in Nearly Linear Time via Path Covers

TL;DR

The paper solves negative-weight SSSP on directed graphs in deterministic near-linear time by introducing path covers, a deterministic analogue of low-diameter decompositions. It defines a projection-based path-cover framework that deterministically covers all short paths with a near-linear-size, $\widetilde{O}(d)$-clustered graph, enabling efficient recursive construction. The main contributions are a deterministic $d$-path-covering projection theorem and a restricted SSSP algorithm that, together, yield a near-linear-time deterministic SSSP for general graphs with running time $O(m\log^{8} n\log(nW))$, improving from prior randomized approaches. The approach derandomizes a broad class of directed-graph SSSP techniques and introduces path covers as a potentially powerful tool for future deterministic algorithms on directed graphs.

Abstract

We present the first deterministic nearly-linear time algorithm for single-source shortest paths with negative edge weights on directed graphs: given a directed graph $G$ with $n$ vertices, $m$ edges whose weights are integer in $\{-W,\dots,W\}$, our algorithm either computes all distances from a source $s$ or reports a negative cycle in time $\tilde{O}(m)\cdot \log(nW)$ time. All known near-linear time algorithms for this problem have been inherently randomized, as they crucially rely on low-diameter decompositions. To overcome this barrier, we introduce a new structural primitive for directed graphs called the path cover. This plays a role analogous to neighborhood covers in undirected graphs, which have long been central to derandomizing algorithms that use low-diameter decomposition in the undirected setting. We believe that path covers will serve as a fundamental tool for the design of future deterministic algorithms on directed graphs.

Figures (2)

• Figure 1: At the top right, $G'$ is a $1$-clustered projection to $G$ that covers all $3$-paths in $G$. At the second row, $G'_1,G'_2,G'_3,G'_4$ are $1$-clustered subgraphs of $G$ that cover all $3$-paths in $G$.
• Figure 2: An illustration of the construction. A path entering $V-\bar{B}^{\mathrm{out}}$ (for example a path starting at $w$) will not leave $B^{\mathrm{out}}$ again because of the length $d$ layer in the middle, thus covered by $H^{\mathrm{out}}$.

Theorems & Definitions (38)

• Theorem 1.1: Deterministic SSSP
• Theorem 1.2: Barrier for Subgraphs
• Theorem 1.3: Covering $d$-Paths with Clustered Projection
• lemma 3.1: Johnson77
• definition 4.1: Graph Projections
• definition 4.2: Path Covering
• definition 4.3: Clustered Graphs
• definition 4.4: Path Cover
• theorem 4.5: Path Cover via Clustered DAGs
• remark 4.6