Universe Reduction for APSP: Equivalence of Three Fine-Grained Hypotheses

TL;DR

This paper proves the equivalence of three prominent hypotheses in fine-grained complexity related to the APSP problem, under certain assumptions, and resolves the complexity of several related graph and matrix problems.

Contribution

It introduces an efficient universe reduction for APSP that establishes the equivalence of three key hypotheses, advancing understanding of their interrelations.

Findings

Proves the equivalence of the APSP, Strong APSP, and Directed Unweighted APSP hypotheses.

Resolves the complexity of problems like Node-Weighted APSP and All-Pairs Bottleneck Paths.

Designs matching lower bounds for several long-standing graph and matrix problems.

Abstract

The APSP Hypothesis states that the All-Pairs Shortest Paths (APSP) problem requires time $n^{3-o(1)}$ on graphs with polynomially bounded integer edge weights. Two increasingly stronger assumptions are the Strong APSP Hypothesis and the Directed Unweighted APSP Hypothesis, which state that the fastest-known APSP algorithms on graphs with small weights and unweighted graphs, respectively, are best-possible. In this paper, we design an efficient universe reduction for APSP, which proves that these three hypotheses are, in fact, equivalent, conditioned on $\omega = 2$ and a plausible additive combinatorics assumption. Along the way, we resolve the fine-grained complexity of many long-standing graph and matrix problems with "intermediate" complexity such as Node-Weighted APSP, All-Pairs Bottleneck Paths, Monotone Min-Plus Product in certain settings, and many others, by designing matching APSP-based lower bounds.