Improved Additive Approximation Algorithms for APSP
Ce Jin, Yael Kirkpatrick, Michał Stawarz, Virginia Vassilevska Williams
TL;DR
This work Addresses the additive-approximate all-pairs shortest paths problem in undirected unweighted graphs by introducing a decomposition-based framework that avoids heavy bounded-difference min-plus machinery and instead uses clusters of constant diameter plus a low-degree remainder. The authors prove two key lemmas—the Decomposition Lemma and a Min-Plus Lemma—and leverage them to obtain faster algorithms for +2-, +4-, and +6-additive APSP, notably achieving a randomized $O(n^{2.22548})$ time for +2-APSP and deterministic $O(n^{2+x/(k+1)})$ times for general +2k-APSP, with concrete improvements for +4 and +6. The approach relies on three ingredients: (i) a structural graph decomposition into small-diameter clusters, (ii) efficient min-plus computations within clusters via standard fast matrix multiplication, and (iii) sparse extensions to the remainder, enabling scalable, near-quadratic performance. These results not only improve specific additive-approximation runtimes but also offer a modular technique potentially useful for other shortest-path problems and for exploring additive vs. multiplicative approximations in APSP.
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 [Dürr, 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.