Epic Fail: Emulators can tolerate polynomially many edge faults for free

TL;DR

This paper demonstrates that certain polynomially sized graph emulators can inherently tolerate a polynomial number of edge faults without increasing their size, revealing a surprising difference from vertex fault tolerance.

Contribution

It introduces the first polynomial fault tolerance results for edge failures in $t$-emulators, showing they can tolerate many faults without size increase for odd $k$.

Findings

Polynomially many edge faults can be tolerated for free in certain emulators.

Constructed a 5-emulator with $O(n^{4/3})$ edges tolerant to $O(n^{2/9})$ edge faults.

Fault tolerance for even $k$ remains an open question.

Abstract

A $t$-emulator of a graph $G$ is a graph $H$ that approximates its pairwise shortest path distances up to multiplicative $t$ error. We study fault tolerant $t$-emulators, under the model recently introduced by Bodwin, Dinitz, and Nazari [ITCS 2022] for vertex failures. In this paper we consider the version for edge failures, and show that they exhibit surprisingly different behavior. In particular, our main result is that, for $(2k-1)$-emulators with $k$ odd, we can tolerate a polynomial number of edge faults for free. For example: for any $n$-node input graph, we construct a $5$-emulator ($k=3$) on $O(n^{4/3})$ edges that is robust to $f = O(n^{2/9})$ edge faults. It is well known that $\Omega(n^{4/3})$ edges are necessary even if the $5$-emulator does not need to tolerate any faults. Thus we pay no extra cost in the size to gain this fault tolerance. We leave open the precise range of free fault tolerance for odd $k$, and whether a similar phenomenon can be proved for even $k$.