The Local Lemma Is Asymptotically Tight for SAT

Abstract

The Local Lemma is a fundamental tool of probabilistic combinatorics and theoretical computer science, yet there are hardly any natural problems known where it provides an asymptotically tight answer. The main theme of our article is to identify several of these problems, among them a couple of widely studied extremal functions related to certain restricted versions of the k -SAT problem, where the Local Lemma does give essentially optimal answers. As our main contribution, we construct unsatisfiable k -CNF formulas where every clause has k distinct literals and every variable appears in at most 2/e + o (1)) 2 k / k clauses. The Lopsided Local Lemma, applied with an assignment of random values according to counterintuitive probabilities, shows that this is asymptotically best possible. The determination of this extremal function is particularly important, as it represents the value where the corresponding k -SAT problem exhibits a complexity hardness jump: From having every instance being a YES-instance it becomes NP-hard just by allowing each variable to occur in one more clause. The construction of our unsatisfiable CNF formulas is based on the binary tree approach of Gebauer [2012], and thus the constructed formulas are in the class MU(1) of minimal unsatisfiable formulas having one more clause than variables. The main novelty of our approach here comes in setting up an appropriate continuous approximation of the problem. This leads us to a differential equation, the solution of which we are able to estimate. The asymptotically optimal binary trees are then obtained through a discretization of this solution. The importance of the binary trees constructed is also underlined by their appearance in many other scenarios. In particular, they give asymptotically precise answers for seemingly unrelated problems like the European Tenure Game introduced by Doerr [2004] and a search problem allowing a limited number of consecutive lies. As yet another consequence, we slightly improve the best-known bounds on the maximum degree and maximum edge-degree of a k -uniform Maker’s win hypergraph in the Neighborhood Conjecture of Beck.