Distributed algorithms for fractional coloring

In this paper we study fractional coloring from the angle of distributed computing. Fractional coloring is the linear relaxation of the classical notion of coloring, and has many applications, in particular in scheduling. It was proved by Hasemann, Hirvonen, Rybicki and Suomela (2016) that for every real $\alpha>1$ and integer $\Delta$, a fractional coloring of total weight at most $\alpha(\Delta+1)$ can be obtained deterministically in a single round in graphs of maximum degree $\Delta$, in the LOCAL model of computation. However, a major issue of this result is that the output of each vertex has unbounded size. Here we prove that even if we impose the more realistic assumption that the output of each vertex has constant size, we can find fractional colorings of total weight arbitrarily close to known tight bounds for the fractional chromatic number in several cases of interest. More precisely, we show that for any fixed $\epsilon > 0$ and $\Delta$, a fractional coloring of total weight at most $\Delta+\epsilon$ can be found in $O(\log^*n)$ rounds in graphs of maximum degree $\Delta$ with no $K_{\Delta+1}$, while finding a fractional coloring of total weight at most $\Delta$ in this case requires $\Omega(\log \log n)$ rounds for randomized algorithms and $\Omega( \log n)$ rounds for deterministic algorithms. We also show how to obtain fractional colorings of total weight at most $2+\epsilon$ in grids of any fixed dimension, for any $\epsilon>0$, in $O(\log^*n)$ rounds. Finally, we prove that in sparse graphs of large girth from any proper minor-closed family we can find a fractional coloring of total weight at most $2+\epsilon$, for any $\epsilon>0$, in $O(\log n)$ rounds.