{
  "id": "1006.0744",
  "version": "v3",
  "published": "2010-06-03T21:14:00.000Z",
  "updated": "2016-04-19T20:01:11.000Z",
  "title": "The Local Lemma is asymptotically tight for SAT",
  "authors": [
    "Heidi Gebauer",
    "Tibor Szabo",
    "Gabor Tardos"
  ],
  "comment": "40 pages, 1 figure",
  "categories": [
    "math.CO",
    "cs.CC",
    "cs.DM"
  ],
  "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 paper 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 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 [16] and thus the constructed formulas are in the class MU(1) of minimal unsatisfiable formulas having one more clauses 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 [9] and a search problem allowing a limited number of consecutive lies.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-08-16T19:21:47.000Z",
      "title": "The Local Lemma Is Tight for SAT",
      "abstract": "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 assignment of random values according to counterintuitive probabilities, shows that our result is asymptotically best possible. The determination of this extremal function is particularly important as it represents the value where the k-SAT problem exhibits its 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 asymptotics of other related extremal functions are also determined. Let l(k) denote the maximum number, such that every k-CNF formula with each clause containing k distinct literals and each clause having a common variable with at most l(k) other clauses, is satisfiable. We establish that the lower bound on l(k) obtained from the Local Lemma is asymptotically optimal, i.e., l(k) = (1/e + o(1))2^{k}. The construction of our unsatisfiable CNF-formulas is based on the binary tree approach of [16] and thus the constructed formulas are in the class MU(1)of minimal unsatisfiable formulas having one more clauses than variables. To obtain the asymptotically optimal binary trees we consider a continuous approximation of the problem, set up a differential equation and estimate its solution. The trees are then obtained through a discretization of this solution. The binary trees constructed also give asymptotically precise answers for seemingly unrelated problems like the European Tenure Game introduced by Doerr [9] and the search problem with bounded number of consecutive lies, considered in a problem of the 2012 IMO contest. As yet another consequence we slightly improve two bounds related to the Neighborhood Conjecture of Beck.",
      "comment": "37 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2016-04-19T20:01:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "G.2.1"
    ],
    "keywords": [
      "local lemma",
      "distinct literals",
      "extremal function",
      "complexity hardness jump",
      "construct unsatisfiable k-cnf formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1006.0744G"
    }
  }
}