Asynchronous Fault-Tolerant Distributed Proper Coloring of Graphs

TL;DR

This paper extends asynchronous distributed graph coloring algorithms to crash-prone networks, providing new upper bounds for colorings and establishing lower bounds that show the limitations of such algorithms in asynchronous crash-prone settings.

Contribution

It introduces an asynchronous crash-tolerant coloring algorithm with improved bounds and proves lower bounds for coloring in crash-prone asynchronous networks, correcting previous results.

Findings

An $O( ext{max degree}^2)$-coloring algorithm in $O( ext{log}^* n)$ rounds.

A reduction showing no $k$-coloring for $k<5$ in certain asynchronous crash-prone models.

Improved coloring for cycles, achieving 5-coloring in $O( ext{log}^* n)$ rounds.

Abstract

We revisit asynchronous computing in networks of crash-prone processes, under the asynchronous variant of the standard LOCAL model, recently introduced by Fraigniaud et al. [DISC 2022]. We focus on the vertex coloring problem, and our contributions concern both lower and upper bounds for this problem. On the upper bound side, we design an algorithm tolerating an arbitrarily large number of crash failures that computes an $O(\Delta^2)$-coloring of any $n$-node graph of maximum degree $\Delta$, in $O(\log^\star n)$ rounds. This extends Linial's seminal result from the (synchronous failure-free) LOCAL model to its asynchronous crash-prone variant. Then, by allowing a dependency on $\Delta$ on the runtime, we show that we can reduce the colors to $\big(\frac12(\Delta+1)(\Delta+2)-1 \big)$. For cycles (i.e., for $\Delta=2$), our algorithm achieves a 5-coloring of any $n$-node cycle, in $O(\log^\star n)$ rounds. This improves the known 6-coloring algorithm by Fraigniaud et al., and fixes a bug in their algorithm, which was erroneously claimed to produce a 5-coloring. On the lower bound side, we show that, for $k<5$, and for every prime integer~$n$, no algorithm can $k$-color the $n$-node cycle in the asynchronous crash-prone variant of LOCAL, independently from the round-complexities of the algorithms. This lower bound is obtained by reduction from an original extension of the impossibility of solving weak symmetry-breaking in the wait-free shared-memory model. We show that this impossibility still holds even if the processes are provided with inputs susceptible to help breaking symmetry.