Distributed Coloring of Graphs with an Optimal Number of Colors

This paper studies sufficient conditions to obtain efficient distributed algorithms coloring graphs optimally (i.e. with the minimum number of colors) in the LOCAL model of computation. Most of the work on distributed vertex coloring so far has focused on coloring graphs of maximum degree Delta with at most Delta+1 colors (or Delta colors when some simple obstructions are forbidden). When Delta is sufficiently large and c >= Delta-k_Delta+1, for some integer k_Delta ~~ sqrt{Delta}-2, we give a distributed algorithm that given a c-colorable graph G of maximum degree Delta, finds a c-coloring of G in min{O((log Delta)^{13/12}log n), 2^{O(log Delta+sqrt{log log n})}} rounds, with high probability. The lower bound Delta-k_Delta+1 is best possible in the sense that for infinitely many values of Delta, we prove that when chi(G) = Delta-k_Delta deciding whether chi(G) <= c is in P, while Embden-Weinert et al. proved that for c <= Delta-k_Delta-1, the same problem is NP-complete. Note that the sequential and distributed thresholds differ by one. Our first result covers the case where the chromatic number of the graph ranges between Delta-sqrt{Delta} and Delta+1. Our second result covers a larger range, but gives a weaker bound on the number of colors: For any sufficiently large Delta, and Omega(log Delta) 0, with a randomized algorithm running in O(log n/log log n) rounds with high probability

Equivariant maps to subshifts whose points have small stabilizers

Let $\Gamma$ be a countably infinite group. Given $k \in \mathbb{N}$, we use $\mathrm{Free}(k^\Gamma)$ to denote the free part of the Bernoulli shift action of $\Gamma$ on $k^\Gamma$. Seward and Tucker-Drob showed that there exists a free subshift $\mathcal{S} \subseteq \mathrm{Free}(2^\Gamma)$ such that every free Borel action of $\Gamma$ on a Polish space admits a Borel $\Gamma$-equivariant map to $\mathcal{S}$. Here we generalize this result as follows. Let $\mathcal{S}$ be a subshift of finite type (for example, $\mathcal{S}$ could be the set of all proper colorings of the Cayley graph of $\Gamma$ with some finite number of colors). Suppose that $\pi \colon \mathrm{Free}(k^\Gamma) \to \mathcal{S}$ is a continuous $\Gamma$-equivariant map and let $\mathrm{Stab}(\pi)$ be the set of all group elements that fix every point in the image of $\pi$. Unless $\pi$ is constant, $\mathrm{Stab}(\pi)$ is a finite normal subgroup of $\Gamma$. We prove that there exists a subshift $\mathcal{S}' \subseteq \mathcal{S}$ such that the stabilizer of every point in $\mathcal{S}'$ is $\mathrm{Stab}(\pi)$ and every free Borel action of $\Gamma$ on a Polish space admits a Borel $\Gamma$-equivariant map to $\mathcal{S}'$. As an application, we deduce that if $F$ is a nonempty finite symmetric subset of $\Gamma$ of size $|F| = d$ not containing the identity and $\mathrm{Col}(F, d + 1) \subseteq (d+1)^\Gamma$ is the set of all proper $(d+1)$-colorings of the Cayley graph of $\Gamma$ corresponding to $F$, then there is a free subshift $\mathcal{S} \subseteq \mathrm{Col}(F, d+1)$ such that every free Borel action of $\Gamma$ on a Polish space admits a Borel $\Gamma$-equivariant map to $\mathcal{S}$.Comment: 22 p

Distributed Algorithms, the Lov\'{a}sz Local Lemma, and Descriptive Combinatorics

In this paper we consider coloring problems on graphs and other combinatorial structures on standard Borel spaces. Our goal is to obtain sufficient conditions under which such colorings can be made well-behaved in the sense of topology or measure. To this end, we show that such well-behaved colorings can be produced using certain powerful techniques from finite combinatorics and computer science. First, we prove that efficient distributed coloring algorithms (on finite graphs) yield well-behaved colorings of Borel graphs of bounded degree; roughly speaking, deterministic algorithms produce Borel colorings, while randomized algorithms give measurable and Baire-measurable colorings. Second, we establish measurable and Baire-measurable versions of the Symmetric Lov\'{a}sz Local Lemma (under the assumption $\mathsf{p}(\mathsf{d}+1)^8 \leq 2^{-15}$, which is stronger than the standard LLL assumption $\mathsf{p}(\mathsf{d} + 1) \leq e^{-1}$ but still sufficient for many applications). From these general results, we derive a number of consequences in descriptive combinatorics and ergodic theory.Comment: 35 page