Asynchronous Fault-Tolerant Distributed Proper Coloring of Graphs

TL;DR

The paper tackles proper vertex coloring in the ASYNC LOCAL model with crash-prone processes, bridging asynchronous crash-prone networks and classical LOCAL colorings. It first adapts Linial’s Δ^2-coloring approach to ASYNC LOCAL to achieve O(Δ^2) colors in O(log^* n) rounds, then introduces a color-reduction technique that yields a (1/2)(Δ+1)(Δ+2) colors plus a further refinement to (1/2)(Δ+1)(Δ+2)−1 colors, maintaining essentially the same asymptotic round complexity with a Δ-dependent additive term. A dedicated refinement leverages special color-pair properties and edge flips to save one color, enabling a 5-coloring of cycles (Δ=2) in O(log^* n) rounds, and clarifies bugs in earlier 5-color cycle algorithms. The work also proves strong lower bounds: for prime n, no algorithm can color the n-cycle with k<5 colors in ASYNC LOCAL, via reductions from weak symmetry breaking in wait-free shared memory with inputs, and extends impossibility results to several coloring and MIS tasks, delineating the limits of ASYNC LOCAL. Overall, the paper advances understanding of symmetry breaking and coloring in asynchronous crash-prone networks, providing tight upper bounds, constructive algorithms, and rigorous lower bounds that shape what is computable in this model.

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 $O(Δ^2)$-coloring of any $n$-node graph of maximum degree $Δ$, 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 $Δ$ on the runtime, we show that we can reduce the colors to $\big(\frac12(Δ+1)(Δ+2)-1 \big)$. For cycles (i.e., for $Δ=2$), 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.

Theorems & Definitions (34)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Theorem 5
• Corollary 5
• Corollary 5
• Corollary 5
• Theorem 6
• Theorem 7