Improved Additive Approximation Algorithms for APSP

TL;DR

This paper presents improved algorithms for approximate all-pairs shortest paths in undirected unweighted graphs, achieving faster runtimes for various approximation factors by employing a novel clustering technique and standard matrix multiplication.

Contribution

The authors introduce a simple clustering-based approach that improves the time complexity of approximate APSP algorithms without relying on bounded-difference matrix multiplication.

Findings

Achieved $O(n^{2.2255})$ time for +2-approximate APSP.

Improved time complexities for +4 and +6-approximate APSP to $O(n^{2.1462})$ and $O(n^{2.1026})$ respectively.

Introduced a clustering technique based on constant diameter clusters and low degree vertices that could be of independent interest.

Abstract

The All-Pairs Shortest Paths (APSP) is a foundational problem in theoretical computer science. Approximating APSP in undirected unweighted graphs has been studied for many years, beginning with the work of Dor, Halperin and Zwick [SICOMP'01]. Many recent works have attempted to improve these original algorithms using the algebraic tools of fast matrix multiplication. We improve on these results for the following problems. For $+2$-approximate APSP, the state-of-the-art algorithm runs in $O(n^{2.259})$ time [D\"urr, IPL 2023; Deng, Kirkpatrick, Rong, Vassilevska Williams, and Zhong, ICALP 2022]. We give an improved algorithm in $O(n^{2.2255})$ time. For $+4$ and $+6$-approximate APSP, we achieve time complexities $O(n^{2.1462})$ and $O(n^{2.1026})$ respectively, improving the previous $O(n^{2.155})$ and $O(n^{2.103})$ achieved by [Saha and Ye, SODA 2024]. In contrast to previous works, we do not use the big hammer of bounded-difference $(\min,+)$-product algorithms. Instead, our algorithms are based on a simple technique that decomposes the input graph into a small number of clusters of constant diameter and a remainder of low degree vertices, which could be of independent interest in the study of shortest paths problems. We then use only standard fast matrix multiplication to obtain our improvements.