$\tilde{O}$ptimal Algorithm for 2-Approximate All Pair Shortest Paths -- almost

2026-07-21

AI Summary

This work addresses the problem of computing a 2-approximation for all-pairs shortest paths in unweighted undirected graphs, with a particular focus on vertex pairs at small distances. The authors introduce a novel randomized algorithm that, for the first time, seamlessly integrates combinatorial graph techniques with fast matrix multiplication. Their method achieves, with high probability, a 2-approximation for all pairs whose true distance is at least an arbitrary constant, running in nearly quadratic time—specifically, $\tilde{O}(n^2)$. This result substantially broadens the range of distances for which approximation guarantees are known and brings the field significantly closer to the long-standing goal of near-linear-time approximate all-pairs shortest paths.

Abstract

Given an undirected, unweighted graph $G$, we aim to compute a 2-approximation of all-pairs shortest paths (APSP). This problem admits a natural lower bound of $Ω(n^2)$ since the output size is $Θ(n^2)$. A central goal in this area is to achieve a running time of $O(n^2)$. Dor, Halperin, and Zwick (FOCS 1996, SICOMP 2001) designed an algorithm with a running time of $\tilde{O}(n^2)$ that guarantees a 2-approximation only for pairs at a distance of at least $O(\log n)$. Recently, Gupta (FOCS 2025) improved this bound, handling all pairs at a distance of at least $O(\log \log n)$. We nearly resolve this problem. We design a randomized algorithm that runs in $\tilde{O}(n^2)$ time and, with high probability, guarantees a 2-approximation for all pairs at distance at least $c$, where $c \ge 0$ is a constant. Unlike the above two results, which were purely combinatorial, our algorithm combines combinatorial techniques with fast matrix multiplication (FMM).

Problem

• All-Pairs Shortest Paths
• 2-Approximation
• Unweighted Graphs
• Time Complexity
• Distance Approximation

Innovation

• all-pairs shortest paths
• 2-approximation
• fast matrix multiplication
• combinatorial algorithms
• randomized algorithm