Improved Additive Approximation Algorithms for APSP

2025-11-06

AI Summary

This paper studies additive approximation algorithms for the All-Pairs Shortest Paths (APSP) problem on undirected unweighted graphs. We propose a novel algorithmic framework based on graph clustering decomposition: the graph is partitioned into clusters of constant diameter and a low-degree set of remaining vertices; intra- and inter-cluster distances are then computed via standard fast matrix multiplication, circumventing reliance on sophisticated (min,+)-convolution techniques. Our approach achieves the best-known time complexities of O(n²·²²⁵⁵), O(n²·¹⁴⁶²), and O(n²·¹⁰²⁶) for +2-, +4-, and +6-additive approximations, respectively. The key innovation lies in replacing high-complexity algebraic machinery with a conceptually simple yet powerful graph decomposition, enabling improved efficiency without sacrificing approximation accuracy. This paradigm shift offers a new design principle for APSP approximation algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

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.

Problem

• Improving additive approximation algorithms for All-Pairs Shortest Paths
• Developing faster algorithms for +2, +4, and +6 approximate APSP
• Using graph decomposition and standard matrix multiplication techniques

Innovation

• Decomposes graph into constant diameter clusters
• Uses standard fast matrix multiplication techniques
• Avoids bounded-difference min-plus product algorithms