### 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 F_{k}(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) = 2^{k}ln 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 F_{k}(n, m = rn) is p-satisfiable tends to 0 as n tends to infinity. We prove that there exists a sequence δ_{k} → 0 such that if r ≤ (1 - δ_{k})τ(p) then the probability that F_{k}(n, m = rn) is p-satisfiable tends to 1 as n tends to infinity. The sequence δ_{k} 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 language | English (US) |
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Title of host publication | Annual Symposium on Foundations of Computer Science - Proceedings |

Pages | 362-370 |

Number of pages | 9 |

State | Published - 2003 |

Event | Proceedings: 44th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2003 - Cambridge, MA, United States Duration: Oct 11 2003 → Oct 14 2003 |

### Other

Other | Proceedings: 44th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2003 |
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Country | United States |

City | Cambridge, MA |

Period | 10/11/03 → 10/14/03 |

### Fingerprint

### ASJC Scopus subject areas

- Hardware and Architecture

### Cite this

*Annual Symposium on Foundations of Computer Science - Proceedings*(pp. 362-370)

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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.

}

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

SP - 362

EP - 370

BT - Annual Symposium on Foundations of Computer Science - Proceedings

ER -