Deterministic coloring algorithms in the LOCAL model

Abstract: We study the problem of bi-chromatic coloring of hypergraphs in the LOCAL distributed model of computation. This problem can easily be solved by a randomized local algorithm with no communication. However, it is not known how to solve it deterministically with only a polylogarithmic number of communication rounds. In this paper we indeed design such a deterministic algorithm that solves this problem with polylogarithmic number of communication rounds. This is an almost exponential improvement on the previously known deterministic local algorithms for this problem. Because the bi-chromatic coloring of hypergraphs problem is known to be complete in the class of all locally checkable graph problems, our result implies deterministic local algorithms with polylogarithmic number of communication rounds for all such problems for which an efficient randomized algorithm exists. This solves one of the fundamental open problems in the area of local distributed graph algorithms. By reductions due to Ghaffari, Kuhn and Maus [STOC 2017] this implies such polylogarithmically efficient deterministic local algorithms for many graph problems.