Fast Sampling and Counting k -SAT Solutions in the Local Lemma Regime

We give new algorithms based on Markov chains to sample and approximately count satisfying assignments to k -uniform CNF formulas where each variable appears at most d times. For any k and d satisfying kd < n o(1) and k ≥ 20 log k + 20 log d + 60, the new sampling algorithm runs in close to linear time, and the counting algorithm runs in close to quadratic time. Our approach is inspired by Moitra (JACM, 2019), which remarkably utilizes the Lovász local lemma in approximate counting. Our main technical contribution is to use the local lemma to bypass the connectivity barrier in traditional Markov chain approaches, which makes the well-developed MCMC method applicable on disconnected state spaces such as SAT solutions. The benefit of our approach is to avoid the enumeration of local structures and obtain fixed polynomial running times, even if k = ω (1) or d = ω (1).

A Meeting Point of Probability, Graphs, and Algorithms: The Lovász Local Lemma and Related Results—A Survey

A classic and fundamental result, known as the Lovász Local Lemma, is a gem in the probabilistic method of combinatorics. At a high level, its core message can be described by the claim that weakly dependent events behave similarly to independent ones. A fascinating feature of this result is that even though it is a purely probabilistic statement, it provides a valuable and versatile tool for proving completely deterministic theorems. The Lovász Local Lemma has found many applications; despite being originally published in 1973, it still attracts active novel research. In this survey paper, we review various forms of the Lemma, as well as some related results and applications.

Majority Colorings of Sparse Digraphs

A majority coloring of a directed graph is a vertex-coloring in which every vertex has the same color as at most half of its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that every digraph has a majority $4$-coloring and conjectured that every digraph admits a majority $3$-coloring. They observed that the Local Lemma implies the conjecture for digraphs of large enough minimum out-degree if, crucially, the maximum in-degree is bounded by a(n exponential) function of the minimum out-degree. Our goal in this paper is to develop alternative methods that allow the verification of the conjecture for natural, broad digraph classes, without any restriction on the in-degrees. Among others, we prove the conjecture 1) for digraphs with chromatic number at most $6$ or dichromatic number at most $3$, and thus for all planar digraphs; and 2) for digraphs with maximum out-degree at most $4$. The benchmark case of $r$-regular digraphs remains open for $r \in [5,143]$. Our inductive proofs depend on loaded inductive statements about precoloring extensions of list-colorings. This approach also gives rise to stronger conclusions, involving the choosability version of majority coloring. We also give further evidence towards the existence of majority-$3$-colorings by showing that every digraph has a fractional majority 3.9602-coloring. Moreover we show that every digraph with large enough minimum out-degree has a fractional majority $(2+\varepsilon)$-coloring.