New versions of Lovász Local Lemma and their applications

Abstract

Lovasz Local Lemma (LLL) is an important tool in combinatorics and probability theory. It can be used to show the existence of combinatorial objects meeting a collection of criteria as long as the criteria are weakly dependent. It was first proposed by ErdHos and Lovasz in 1975. Since then, many applications of LLL have been found in combinatorics, theoretical computer science, and physics. Recently, several new versions of LLL have been proposed. Constructive LLL is an especially big breakthrough in theoretical computer science that has attracted lots of attention. In this paper, we will review recent progress in LLL research, including new versions of LLL and their applications. We will precisely define and differentiate among abstract LLL, lopsided LLL, variable LLL, and quantum LLL. We will also provide connections between abstract LLL and statistical physics, as well as between quantum LLL and quantum physics. LLL can be used to prove the existence of solutions, find solutions efficiently, count the number of solutions, and sample a solution uniformly at random. We will also illustrate these applications of LLL with the SAT problem and the quantum SAT problem.