Faster Approximate All Pairs Shortest Paths

TL;DR

This paper introduces new algorithms for faster approximate solutions to the all pairs shortest path problem in undirected graphs, improving existing bounds for both multiplicative and additive approximations using combinatorial and fast matrix multiplication techniques.

Contribution

It presents the first improved algorithms for multiplicative 2-approximation in unweighted graphs and enhances bounds for additive approximations, highlighting the role of fast matrix multiplication in approximation.

Findings

Deterministic 2-approximation runs in O(n^{2.072}) time.

Randomized 2-approximation runs in O(n^{2.032}) time, improving previous bounds.

New combinatorial and FMM-based algorithms for approximating paths of length  k for dense graphs.

Abstract

The all pairs shortest path problem (APSP) is one of the foundational problems in computer science. For weighted dense graphs on $n$ vertices, no truly sub-cubic algorithms exist to compute APSP exactly even for undirected graphs. This is popularly known as the APSP conjecture and has played a prominent role in developing the field of fine-grained complexity. The seminal result of Seidel uses fast matrix multiplication (FMM) to compute APSP on unweighted undirected graphs exactly in $\tilde{O}(n^{\omega})$ time, where $\omega=2.372$. Even for unweighted undirected graphs, it is not possible to obtain a $(2-\epsilon)$-approximation of APSP in $o(n^\omega)$ time. In this paper, we provide a multitude of new results for multiplicative and additive approximations of APSP in undirected graphs for both unweighted and weighted cases. We provide new algorithms for multiplicative 2-approximation of unweighted graphs: a deterministic one that runs in $\tilde{O}(n^{2.072})$ time and a randomized one that runs in $\tilde{O}(n^{2.032})$ on expectation improving upon the best known bound of $\tilde{O}(n^{2.25})$ by Roditty (STOC, 2023). For $2$-approximating paths of length $\geq k$, $k \geq 4$, we provide the first improvement after Dor, Halperin, Zwick (2000) for dense graphs even just using combinatorial methods, and then improve it further using FMM. We next consider additive approximations, and provide improved bounds for all additive $\beta$-approximations, $\beta \geq 4$. For weighted graphs, we show that by allowing small additive errors along with an $(1+\epsilon)$-multiplicative approximation, it is possible to improve upon Zwick's $\tilde{O}(n^\omega)$ algorithm. Our results point out the crucial role that FMM can play even on approximating APSP on unweighted undirected graphs, and reveal new bottlenecks towards achieving a quadratic running time to approximate APSP.