Deterministic Distributed Vertex Coloring: Simpler, Faster, and without Network Decomposition

We present a simple deterministic distributed algorithm that computes a $(\Delta+1)$-vertex coloring in $O(\log^2 \Delta \cdot \log n)$ rounds, in any graph with at most $n$ nodes and maximum degree at most $\Delta$. The algorithm can be implemented with $O(\log n)$-bit messages. It can also be extended to the more general $(degree+1)$-list coloring problem. Obtaining a polylogarithmic-time deterministic algorithm for $(\Delta+1)$-vertex coloring had remained a central open question in the area of distributed graph algorithms since the 1980s, until a recent network decomposition algorithm of Rozho\v{n} and Ghaffari [STOC '20]. The current state of the art is based on an improved variant of their decomposition, which leads to an $O(\log^5 n)$-round algorithm for $(\Delta+1)$-vertex coloring. Our coloring algorithm is completely different and considerably simpler and faster. It solves the coloring problem in a direct way, without using network decomposition, by gradually rounding a certain fractional color assignment until reaching an integral color assignment. Moreover, via the approach of Chang, Li, and Pettie [STOC '18], this improved deterministic algorithm also leads to an improvement in the complexity of randomized algorithms for $(\Delta+1)$-coloring, now reaching the bound of $O(\log^3\log n)$ rounds.