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
$O(\text{log}^{2}\Delta. \text{log}\ n)$
rounds. The algorithm can be implemented with
$O(\text{log}\ n)$
-bit messages. The algorithm 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 Rozhoň and Ghaffari [STOC'20]. The current state of the art is based on an improved variant of their decomposition, which leads to an
$O(\text{log}^{5}n)$
-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 assignments. 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
$O(\text{log}^{3}\text{log}\ n)$
rounds. As a further application, we provide faster deterministic distributed algorithms for the following vertex coloring variants. In graphs of arboricity
$a$
, we show that a
$(2+\varepsilon)a$
-vertex coloring can be computed in
$O(\text{log}^{3}a\cdot \text{log} n)$
rounds. We also show that for
$\Delta\geq 3$
, a
$\Delta$
-coloring of a
$\Delta$
-colorable graph
$G$
can be computed in
$O(\text{log}^{2}\Delta\cdot \text{log}^{2}n)$
rounds.