Distributed algorithms for fractional coloring

TL;DR

This paper develops distributed algorithms for fractional graph coloring with constant output size, achieving near-optimal total weights in various graph classes within polylogarithmic rounds.

Contribution

It introduces distributed algorithms that find fractional colorings with constant-sized outputs close to tight bounds, improving upon previous unbounded output solutions.

Findings

Fractional coloring of weight at most Δ+ε in graphs with no K_{Δ+1} in O(log* n) rounds.

Lower bounds show that achieving weight Δ in such graphs requires Ω(log log n) randomized and Ω(log n) deterministic rounds.

In fixed-dimension grids, fractional colorings of weight 2+ε are achievable in O(log* n) rounds.

Abstract

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.