Improved Approximation Algorithms for n-Pairs Shortest Paths

2026-07-02

AI Summary

This work addresses the problem of approximately computing shortest-path distances for a given set of $n$ vertex pairs in weighted undirected graphs. The authors propose a novel algorithmic framework based on graph decomposition and careful handling of “heavy edges.” By transforming the dependence on the maximum edge weight along paths into a purely multiplicative approximation, the method achieves—for the first time in non-ultrasparse graphs—an approximation ratio better than $(2 - \alpha)k$. The algorithm runs in $\tilde{O}(mn^{1/k} + n^{1+2/k})$ time and attains a $1.622k$-approximation in weighted graphs, significantly improving upon the previous best-known $(2k - 3)$ bound. It also yields superior time–approximation trade-offs in both unweighted and dense weighted graphs.

Abstract

Let $G = (V, E)$ be a graph with $n = |V|$ nodes and $m = |E|$ edges. The $t$-Pairs Shortest Paths problem, introduced by Cohen [FOCS'93; SICOMP'99], asks to approximate the distances between $t$ prespecified pairs of vertices. Recently, this problem has received renewed attention, particularly in the case where $t = Θ(n)$: the $n$-Pairs Shortest Paths problem. In this setting, new algorithms and conditional lower bounds have been developed by Dalirrooyfard, Jin, Vassilevska Williams, and Wein [FOCS'22], and Chechik, Hoch, and Lifshitz [SODA'25]. In this paper, we present the first algorithm for the $n$-Pairs Shortest Paths problem in \textit{weighted} undirected graphs that achieves a $(2 - α)k$-approximation, for constant $α> 0$, that runs in $\tilde{O}(mn^{1/k} + n^{1 + 2/k})$ time. Specifically, we present a $1.622k$-approximation, improving upon the $(2k - 3)$-approximation of Chechik, Hoch, and Lifshitz [SODA'25] for graphs that are not super sparse, which answers in the affirmative the open question posed by them. We also develop improved approximation algorithms with better tradeoffs for unweighted graphs and dense weighted graphs that improve upon the results of Dalirrooyfard \etal~and Chechik, Hoch, and Lifshitz. Our main technical contribution is the new \textit{heavy-edge} technique. Using this technique, we transform an algorithm with an approximation guarantee that depends on $W_{uv}$, the weight of the heaviest edge on the shortest path between $u$ and $v$, into an algorithm with purely multiplicative approximation that does not depend on $W_{uv}$.

Problem

• n-Pairs Shortest Paths
• approximation algorithms
• weighted graphs
• shortest path distance
• graph algorithms

Innovation

• n-Pairs Shortest Paths
• approximation algorithms
• heavy-edge technique
• weighted undirected graphs
• multiplicative approximation