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

2026-03-29

AI Summary

This work investigates whether the standard, strong, and directed unweighted variants of the All-Pairs Shortest Paths (APSP) hypothesis are equivalent under fine-grained complexity assumptions, and clarifies the exact computational limits of several graph and matrix problems. By designing efficient APSP-complete reductions, the paper establishes for the first time the equivalence of these three hypotheses under the assumption that the matrix multiplication exponent ω equals 2 and a plausible additive combinatorics conjecture holds. This result unifies central conjectures in fine-grained complexity theory and yields matching APSP-based conditional lower bounds for long-standing open problems such as Node-Weighted APSP and All-Pairs Bottleneck Paths, substantially advancing the theoretical understanding of their computational hardness.

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 $ω= 2$ 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.

Problem

• APSP Hypothesis
• Strong APSP Hypothesis
• Directed Unweighted APSP Hypothesis
• fine-grained complexity
• universe reduction

Innovation

• Universe Reduction
• APSP Hypothesis
• Fine-Grained Complexity
• Equivalence of Hypotheses
• Min-Plus Product