Pairwise Reachability Oracles and Preservers under Failures

TL;DR

This paper introduces the first efficient fault-tolerant reachability oracles and preservers for arbitrary node pairs in directed graphs, capable of handling multiple failures with optimal size bounds and constant query time.

Contribution

It presents novel constructions of dual fault-tolerant reachability oracles and preservers for arbitrary pairs, surpassing previous bounds based on single-source scenarios.

Findings

First non-trivial dual fault-tolerant reachability oracles with constant query time.

Extremal bounds for sparse reachability preservers resilient to multiple failures.

Affirmative answer to whether size bounds can be improved beyond single-source derived bounds.

Abstract

In this paper, we consider reachability oracles and reachability preservers for directed graphs/networks prone to edge/node failures. Let $G = (V, E)$ be a directed graph on $n$-nodes, and $P\subseteq V\times V$ be a set of vertex pairs in $G$. We present the first non-trivial constructions of single and dual fault-tolerant pairwise reachability oracle with constant query time. Furthermore, we provide extremal bounds for sparse fault-tolerant reachability preservers, resilient to two or more failures. Prior to this work, such oracles and reachability preservers were widely studied for the special scenario of single-source and all-pairs settings. However, for the scenario of arbitrary pairs, no prior (non-trivial) results were known for dual (or more) failures, except those implied from the single-source setting. One of the main questions is whether it is possible to beat the $O(n |P|)$ size bound (derived from the single-source setting) for reachability oracle and preserver for dual failures (or $O(2^k n|P|)$ bound for $k$ failures). We answer this question affirmatively.