A Constant-Approximation Distance Labeling Scheme under Polynomially Many Edge Failures

2026-04-02

AI Summary

This work addresses the problem of efficiently approximating distances in undirected weighted graphs under an arbitrary polynomial number of edge failures. The authors present the first deterministic polynomial-time fault-tolerant distance labeling scheme that, given labels of the query endpoints and the failed edges, achieves a constant-factor approximation when up to $f$ edges fail. The constructed labels have size $\widetilde{O}(f^4 n^{1/k})$ and yield an approximation ratio of $O(k^4)$, thereby resolving an open question posed by Dory and Parter and overcoming the prior limitation of approximation ratios growing linearly with $f$. In applications such as distance-sensitive oracles, the scheme maintains sublinear query time and bounded approximation error even when $f = \Theta(\log n)$, significantly outperforming existing approaches.

Abstract

A fault-tolerant distance labeling scheme assigns a label to each vertex and edge of an undirected weighted graph $G$ with $n$ vertices so that, for any edge set $F$ of size $|F| \leq f$, one can approximate the distance between $p$ and $q$ in $G \setminus F$ by reading only the labels of $F \cup \{p,q\}$. For any $k$, we present a deterministic polynomial-time scheme with $O(k^{4})$ approximation and $\tilde{O}(f^{4}n^{1/k})$ label size. This is the first scheme to achieve a constant approximation while handling any number of edge faults $f$, resolving the open problem posed by Dory and Parter [DP21]. All previous schemes provided only a linear-in-$f$ approximation [DP21, LPS25]. Our labeling scheme directly improves the state of the art in the simpler setting of distance sensitivity oracles. Even for just $f = Θ(\log n)$ faults, all previous oracles either have super-linear query time, linear-in-$f$ approximation [CLPR12], or exponentially worse $2^{{\rm poly}(k)}$ approximation dependency in $k$ [HLS24].

Problem

• fault-tolerant
• distance labeling
• edge failures
• constant approximation
• graph algorithms

Innovation

• fault-tolerant labeling
• constant approximation
• distance sensitivity oracle
• edge failures
• polynomial-time scheme