Deterministic Distributed Vertex Coloring in Polylogarithmic Time

Consider an n-vertex graph G = (V,E) of maximum degree Delta, and suppose that each vertex v \in V hosts a processor. The processors are allowed to communicate only with their neighbors in G. The communication is synchronous, i.e., it proceeds in discrete rounds. In the distributed vertex coloring problem the objective is to color G with Delta + 1, or slightly more than Delta + 1, colors using as few rounds of communication as possible. (The number of rounds of communication will be henceforth referred to as running time.) Efficient randomized algorithms for this problem are known for more than twenty years \cite{L86, ABI86}. Specifically, these algorithms produce a (Delta + 1)-coloring within O(log n) time, with high probability. On the other hand, the best known deterministic algorithm that requires polylogarithmic time employs O(Delta²⁾ colors. This algorithm was devised in a seminal FOCS'87 paper by Linial \cite{L87}. Its running time is O(log^* n). In the same paper Linial asked whether one can color with significantly less than Delta² colors in deterministic polylogarithmic time. By now this question of Linial became one of the most central long-standing open questions in this area. In this paper we answer this question in the affirmative, and devise a deterministic algorithm that employs \Delta^{1 +o(1)} colors, and runs in polylogarithmic time. Specifically, the running time of our algorithm is O(f(Delta) log Delta log n), for an arbitrarily slow-growing function f(Delta) = \omega(1). We can also produce O(Delta^{1 + \eta})-coloring in O(log Delta log n)-time, for an arbitrarily small constant \eta > 0, and O(Delta)-coloring in O(Delta^{\epsilon} log n) time, for an arbitrarily small constant \epsilon > 0.