On the maximum satisfiability of random formulas

Dimitris Achlioptas, Assaf Naor, Yuval Peres

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Maximum satisfiability is a canonical NP-complete problem that appears empirically hard for random instances. At the same time, it is rapidly becoming a canonical problem for statistical physics. In both of these realms, evaluating new ideas relies crucially on knowing the maximum number of clauses one can typically satisfy in a random k-CNF formula. In this paper we give asymptotically tight estimates for this quantity. Let us say that a k-CNF formula is p-satisfiable if there exists a truth assignment satisfying 1-2-k +p2-k fraction of all clauses (every k-CNF is 0-satisfiable). Let Fk(n, m) denote a random k-CNF formula on n variables formed by selecting uniformly, independently and with replacement m out of all (2n)k possible k-clauses. Finally, let τ(p) = 2kln 2/(p + (1 - p) ln(1 - p)). It is easy to prove that for every k ≥ 2 and p ∈ (0, 1), if r ≥ τ (p) then the probability that Fk(n, m = rn) is p-satisfiable tends to 0 as n tends to infinity. We prove that there exists a sequence δk → 0 such that if r ≤ (1 - δk)τ(p) then the probability that Fk(n, m = rn) is p-satisfiable tends to 1 as n tends to infinity. The sequence δk tends to 0 exponentially fast in k. Indeed, even for moderate values of k, e.g. k = 10, our result gives very tight bounds for the fraction of satisfiable clauses in a random k-CNF. In particular, for k > 2 it improves upon all previously known such bound.

Original languageEnglish (US)
Title of host publicationAnnual Symposium on Foundations of Computer Science - Proceedings
Pages362-370
Number of pages9
StatePublished - 2003
EventProceedings: 44th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2003 - Cambridge, MA, United States
Duration: Oct 11 2003Oct 14 2003

Other

OtherProceedings: 44th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2003
CountryUnited States
CityCambridge, MA
Period10/11/0310/14/03

Fingerprint

Computational complexity
Physics

ASJC Scopus subject areas

  • Hardware and Architecture

Cite this

Achlioptas, D., Naor, A., & Peres, Y. (2003). On the maximum satisfiability of random formulas. In Annual Symposium on Foundations of Computer Science - Proceedings (pp. 362-370)

On the maximum satisfiability of random formulas. / Achlioptas, Dimitris; Naor, Assaf; Peres, Yuval.

Annual Symposium on Foundations of Computer Science - Proceedings. 2003. p. 362-370.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Achlioptas, D, Naor, A & Peres, Y 2003, On the maximum satisfiability of random formulas. in Annual Symposium on Foundations of Computer Science - Proceedings. pp. 362-370, Proceedings: 44th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2003, Cambridge, MA, United States, 10/11/03.
Achlioptas D, Naor A, Peres Y. On the maximum satisfiability of random formulas. In Annual Symposium on Foundations of Computer Science - Proceedings. 2003. p. 362-370
Achlioptas, Dimitris ; Naor, Assaf ; Peres, Yuval. / On the maximum satisfiability of random formulas. Annual Symposium on Foundations of Computer Science - Proceedings. 2003. pp. 362-370
@inproceedings{bf812bbf10304a5982a9d13ec36b3d12,
title = "On the maximum satisfiability of random formulas",
abstract = "Maximum satisfiability is a canonical NP-complete problem that appears empirically hard for random instances. At the same time, it is rapidly becoming a canonical problem for statistical physics. In both of these realms, evaluating new ideas relies crucially on knowing the maximum number of clauses one can typically satisfy in a random k-CNF formula. In this paper we give asymptotically tight estimates for this quantity. Let us say that a k-CNF formula is p-satisfiable if there exists a truth assignment satisfying 1-2-k +p2-k fraction of all clauses (every k-CNF is 0-satisfiable). Let Fk(n, m) denote a random k-CNF formula on n variables formed by selecting uniformly, independently and with replacement m out of all (2n)k possible k-clauses. Finally, let τ(p) = 2kln 2/(p + (1 - p) ln(1 - p)). It is easy to prove that for every k ≥ 2 and p ∈ (0, 1), if r ≥ τ (p) then the probability that Fk(n, m = rn) is p-satisfiable tends to 0 as n tends to infinity. We prove that there exists a sequence δk → 0 such that if r ≤ (1 - δk)τ(p) then the probability that Fk(n, m = rn) is p-satisfiable tends to 1 as n tends to infinity. The sequence δk tends to 0 exponentially fast in k. Indeed, even for moderate values of k, e.g. k = 10, our result gives very tight bounds for the fraction of satisfiable clauses in a random k-CNF. In particular, for k > 2 it improves upon all previously known such bound.",
author = "Dimitris Achlioptas and Assaf Naor and Yuval Peres",
year = "2003",
language = "English (US)",
pages = "362--370",
booktitle = "Annual Symposium on Foundations of Computer Science - Proceedings",

}

TY - GEN

T1 - On the maximum satisfiability of random formulas

AU - Achlioptas, Dimitris

AU - Naor, Assaf

AU - Peres, Yuval

PY - 2003

Y1 - 2003

N2 - Maximum satisfiability is a canonical NP-complete problem that appears empirically hard for random instances. At the same time, it is rapidly becoming a canonical problem for statistical physics. In both of these realms, evaluating new ideas relies crucially on knowing the maximum number of clauses one can typically satisfy in a random k-CNF formula. In this paper we give asymptotically tight estimates for this quantity. Let us say that a k-CNF formula is p-satisfiable if there exists a truth assignment satisfying 1-2-k +p2-k fraction of all clauses (every k-CNF is 0-satisfiable). Let Fk(n, m) denote a random k-CNF formula on n variables formed by selecting uniformly, independently and with replacement m out of all (2n)k possible k-clauses. Finally, let τ(p) = 2kln 2/(p + (1 - p) ln(1 - p)). It is easy to prove that for every k ≥ 2 and p ∈ (0, 1), if r ≥ τ (p) then the probability that Fk(n, m = rn) is p-satisfiable tends to 0 as n tends to infinity. We prove that there exists a sequence δk → 0 such that if r ≤ (1 - δk)τ(p) then the probability that Fk(n, m = rn) is p-satisfiable tends to 1 as n tends to infinity. The sequence δk tends to 0 exponentially fast in k. Indeed, even for moderate values of k, e.g. k = 10, our result gives very tight bounds for the fraction of satisfiable clauses in a random k-CNF. In particular, for k > 2 it improves upon all previously known such bound.

AB - Maximum satisfiability is a canonical NP-complete problem that appears empirically hard for random instances. At the same time, it is rapidly becoming a canonical problem for statistical physics. In both of these realms, evaluating new ideas relies crucially on knowing the maximum number of clauses one can typically satisfy in a random k-CNF formula. In this paper we give asymptotically tight estimates for this quantity. Let us say that a k-CNF formula is p-satisfiable if there exists a truth assignment satisfying 1-2-k +p2-k fraction of all clauses (every k-CNF is 0-satisfiable). Let Fk(n, m) denote a random k-CNF formula on n variables formed by selecting uniformly, independently and with replacement m out of all (2n)k possible k-clauses. Finally, let τ(p) = 2kln 2/(p + (1 - p) ln(1 - p)). It is easy to prove that for every k ≥ 2 and p ∈ (0, 1), if r ≥ τ (p) then the probability that Fk(n, m = rn) is p-satisfiable tends to 0 as n tends to infinity. We prove that there exists a sequence δk → 0 such that if r ≤ (1 - δk)τ(p) then the probability that Fk(n, m = rn) is p-satisfiable tends to 1 as n tends to infinity. The sequence δk tends to 0 exponentially fast in k. Indeed, even for moderate values of k, e.g. k = 10, our result gives very tight bounds for the fraction of satisfiable clauses in a random k-CNF. In particular, for k > 2 it improves upon all previously known such bound.

UR - http://www.scopus.com/inward/record.url?scp=0345412679&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0345412679&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0345412679

SP - 362

EP - 370

BT - Annual Symposium on Foundations of Computer Science - Proceedings

ER -