An Optimal Distributed $(Δ+1)$-Coloring Algorithm?

Vertex coloring is one of the classic symmetry breaking problems studied in distributed computing. In this paper we present a new algorithm for $(\Delta+1)$-list coloring in the randomized $\textsf{LOCAL}$ model running in $O(\log^\ast n + \textsf{Det}_d(\text{poly} \log n))$ time, where $\textsf{Det}_d(n')$ is the deterministic complexity of $(\text{deg}+1)$-list coloring ($v$'s palette has size $\text{deg}(v)+1$) on $n'$-vertex graphs. This improves upon a previous randomized algorithm of Harris, Schneider, and Su (STOC 2016). with complexity $O(\sqrt{\log \Delta} + \log\log n + \textsf{Det}_d(\text{poly} \log n))$, and is dramatically faster than the best known deterministic algorithm of Fraigniaud, Heinrich, and Kosowski (FOCS 2016), with complexity $O(\sqrt{\Delta}\log^{2.5}\Delta + \log^* n)$. Our algorithm appears to be optimal. It matches the $\Omega(\log^\ast n)$ randomized lower bound, due to Naor (SIDMA 1991) and sort of matches the $\Omega(\textsf{Det}(\text{poly} \log n))$ randomized lower bound due to Chang, Kopelowitz, and Pettie (FOCS 2016), where $\textsf{Det}$ is the deterministic complexity of $(\Delta+1)$-list coloring. The best known upper bounds on $\textsf{Det}_d(n')$ and $\textsf{Det}(n')$ are both $2^{O(\sqrt{\log n'})}$ by Panconesi and Srinivasan (Journal of Algorithms 1996), and it is quite plausible that the complexities of both problems are the same, asymptotically.