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

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 $Ω(n<sup>2)$ since the output size is $Θ(n<sup>2)$. A central goal in this area is to achieve a running time of $O(n<sup>2)$. Dor, Halperin, and Zwick (FOCS 1996, SICOMP 2001) designed an algorithm with a running time of $\tilde{O}(n<sup>2)$ 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 $\tilde{O}(n<sup>2)$ 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).