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

TL;DR

This paper introduces a novel fault-tolerant distance labeling scheme that provides a constant approximation for any number of edge failures, significantly improving over previous linear-in-f approximation methods.

Contribution

It presents the first deterministic polynomial-time scheme achieving constant approximation with polynomially many edge failures, resolving an open problem in fault-tolerant graph algorithms.

Findings

Achieves $O(k^{4})$ approximation with $ ilde{O}(f^{4}n^{1/k})$ label size.

First scheme to handle any number of edge faults with constant approximation.

Improves the state of the art in distance sensitivity oracles for multiple edge failures.

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 $f = \Theta(\log n)$ 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].