Occupancy fraction, fractional colouring, and triangle fraction

Given $\varepsilon>0$, there exists $f_0$ such that, if $f_0 \le f \le \Delta^2+1$, then for any graph $G$ on $n$ vertices of maximum degree $\Delta$ in which the neighbourhood of every vertex in $G$ spans at most $\Delta^2/f$ edges, (i) an independent set of $G$ drawn uniformly at random has at least $(1/2-\varepsilon)(n/\Delta)\log f$ vertices in expectation, and (ii) the fractional chromatic number of $G$ is at most $(2+\varepsilon)\Delta/\log f$. These bounds cannot in general be improved by more than a factor $2$ asymptotically. One may view these as stronger versions of results of Ajtai, Koml\'os and Szemer\'edi (1981) and Shearer (1983). The proofs use a tight analysis of the hard-core model.Comment: 10 page

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