An Optimal Distributed $(\Delta+1)$-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 $textsf{Det}_d(nu0027)$ is the deterministic complexity of $(text{deg}+1)$-list coloring ($v$u0027s palette has size $text{deg}(v)+1$) on $nu0027$-vertex graphs. This improves upon a previous randomized algorithm of Harris, Schneider, and Su (STOC 2016). with complexity $O(sqrt{log Delta} + loglog n + textsf{Det}_d(text{poly} log n))$, 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 $textsf{Det}_d(nu0027)$ and $textsf{Det}(nu0027)$ are both $2^{O(sqrt{log nu0027})}$ by Panconesi and Srinivasan (Journal of Algorithms 1996), and it is quite plausible that the complexities of both problems are the same, asymptotically.