### The Local Lemma Is Asymptotically Tight for SATThe Local Lemma Is Asymptotically Tight for SAT

Access Restriction
Subscribed

 Author Gebauer, Heidi ♦ Szab, Tibor ♦ Tardos, Gbor Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2016 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Local lemma ♦ Extremal combinatorics ♦ Satisfiability of k-CNF formulas 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 $\textit{k}-SAT$ problem, where the Local Lemma does give essentially optimal answers. As our main contribution, we construct unsatisfiable $\textit{k}-CNF$ formulas where every clause has $\textit{k}$ 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 $\textit{k}-SAT$ 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 $\textit{k}-uniform$ Maker’s win hypergraph in the Neighborhood Conjecture of Beck. Description Author Affiliation: Zurich University of Applied Sciences (Gebauer, Heidi); Freie Universität Berlin, Berlin, Germany (Szab, Tibor); Rényi Institute, Budapest, Hungary (Tardos, Gbor) ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2016-12-06 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 63 Issue Number 5 Page Count 32 Starting Page 1 Ending Page 32

#### Open content in new tab

Source: ACM Digital Library