Simple Local Computation Algorithms for the General Lovasz Local Lemma

We consider the task of designing Local Computation Algorithms (LCA) for applications of the Lov\'{a}sz Local Lemma (LLL). LCA is a class of sublinear algorithms proposed by Rubinfeld et al.~\cite{Ronitt} that have received a lot of attention in recent years. The LLL is an existential, sufficient condition for a collection of sets to have non-empty intersection (in applications, often, each set comprises all objects having a certain property). The ground-breaking algorithm of Moser and Tardos~\cite{MT} made the LLL fully constructive, following earlier results by Beck~\cite{beck_lll} and Alon~\cite{alon_lll} giving algorithms under significantly stronger LLL-like conditions. LCAs under those stronger conditions were given in~\cite{Ronitt}, where it was asked if the Moser-Tardos algorithm can be used to design LCAs under the standard LLL condition. The main contribution of this paper is to answer this question affirmatively. In fact, our techniques yield LCAs for settings beyond the standard LLL condition

Improved bounds on coloring of graphs

Given a graph $G$ with maximum degree $\Delta\ge 3$, we prove that the acyclic edge chromatic number $a'(G)$ of $G$ is such that $a'(G)\le\lceil 9.62 (\Delta-1)\rceil$. Moreover we prove that: $a'(G)\le \lceil 6.42(\Delta-1)\rceil$ if $G$ has girth $g\ge 5\,$; a'(G)\le \lceil5.77 (\Delta-1)\rc if $G$ has girth $g\ge 7$; a'(G)\le \lc4.52(\D-1)\rc if $g\ge 53$; a'(G)\le \D+2\, if g\ge \lceil25.84\D\log\D(1+ 4.1/\log\D)\rceil. We further prove that the acyclic (vertex) chromatic number $a(G)$ of $G$ is such that a(G)\le \lc 6.59 \Delta^{4/3}+3.3\D\rc. We also prove that the star-chromatic number $\chi_s(G)$ of $G$ is such that \chi_s(G)\le \lc4.34\Delta^{3/2}+ 1.5\D\rc. We finally prove that the \b-frugal chromatic number \chi^\b(G) of $G$ is such that \chi^\b(G)\le \lc\max\{k_1(\b)\D,\; k_2(\b){\D^{1+1/\b}/ (\b!)^{1/\b}}\}\rc, where k_1(\b) and k_2(\b) are decreasing functions of \b such that k_1(\b)\in[4, 6] and k_2(\b)\in[2,5]. To obtain these results we use an improved version of the Lov\'asz Local Lemma due to Bissacot, Fern\'andez, Procacci and Scoppola \cite{BFPS}.Comment: Introduction revised. Added references. Corrected typos. Proof of Theorem 2 (items c-f) written in more detail

Extremal bipartite independence number and balanced coloring

In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, we show that every sufficiently large bipartite graph with average degree $\Delta$ and with $n$ vertices on each side has a balanced independent set containing $(1-\epsilon) \frac{\log \Delta}{\Delta} n$ vertices from each side for small $\epsilon > 0$. Secondly, we prove that the vertex set of every sufficiently large balanced bipartite graph with maximum degree at most $\Delta$ can be partitioned into $(1+\epsilon)\frac{\Delta}{\log \Delta}$ balanced independent sets. Both of these results are algorithmic and best possible up to a factor of 2, which might be hard to improve as evidenced by the phenomenon known as `algorithmic barrier' in the literature. The first result improves a recent theorem of Axenovich, Sereni, Snyder, and Weber in a slightly more general setting. The second result improves a theorem of Feige and Kogan about coloring balanced bipartite graphs.Comment: minor change