The normalized autocorrelation length of random max r-Sat converges in probability to (1 − 1/2r)/r

Daniel Berend; Yochai Twitto

doi:10.1007/978-3-319-40970-2_5

The normalized autocorrelation length of random max r-Sat converges in probability to (1 − 1/2^r)/r

Daniel Berend, Yochai Twitto

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

3 Scopus citations

Abstract

In this paper we show that the so-called normalized auto- correlation length of random Max r-Sat converges in probability to (1 − 1/2^r)/r, where r is the number of literals in a clause. We also show that the correlation between the numbers of clauses satisfied by a random pair of assignments of distance d = cn, 0 ≤ c ≤ 1, converges in probability to ((1 − c)^r − 1/2^r)/(1 − 1/2^r). The former quantity is of interest in the area of landscape analysis as a way to better understand problems and assess their hardness for local search heuristics. In [34], it has been shown that it may be calculated in polynomial time for any instance, and its mean value over all instances was discussed. Our results are based on a study of the variance of the number of clauses satisfied by a random assignment, and the covariance of the numbers of clauses satisfied by a random pair of assignments of an arbitrary distance. As part of this study, closed-form formulas for the expected value and vari- ance of the latter two quantities are provided. Note that all results are relevant to random r-Sat as well.

Original language	English
Title of host publication	Theory and Applications of Satisfiability Testing – SAT 2016 - 19th International Conference, Proceedings
Editors	Daniel Le Berre, Nadia Creignou
Publisher	Springer Verlag
Pages	60-76
Number of pages	17
ISBN (Print)	9783319409696
DOIs	https://doi.org/10.1007/978-3-319-40970-2_5
State	Published - 1 Jan 2016
Event	19th International Conference on Theory and Applications of Satisfiability Testing, SAT 2016 - Bordeaux, France Duration: 5 Jul 2016 → 8 Jul 2016

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	9710
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	19th International Conference on Theory and Applications of Satisfiability Testing, SAT 2016
Country/Territory	France
City	Bordeaux
Period	5/07/16 → 8/07/16

Keywords

Autocorrelation length
Combinatorial
Fitness landscapes
Local search
Max sat
Optimization

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/978-3-319-40970-2_5

Cite this

Berend, D., & Twitto, Y. (2016). The normalized autocorrelation length of random max r-Sat converges in probability to (1 − 1/2^r)/r. In D. Le Berre, & N. Creignou (Eds.), Theory and Applications of Satisfiability Testing – SAT 2016 - 19th International Conference, Proceedings (pp. 60-76). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9710). Springer Verlag. https://doi.org/10.1007/978-3-319-40970-2_5

Berend, Daniel ; Twitto, Yochai. / The normalized autocorrelation length of random max r-Sat converges in probability to (1 − 1/2^r)/r. Theory and Applications of Satisfiability Testing – SAT 2016 - 19th International Conference, Proceedings. editor / Daniel Le Berre ; Nadia Creignou. Springer Verlag, 2016. pp. 60-76 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "In this paper we show that the so-called normalized auto- correlation length of random Max r-Sat converges in probability to (1 − 1/2r)/r, where r is the number of literals in a clause. We also show that the correlation between the numbers of clauses satisfied by a random pair of assignments of distance d = cn, 0 ≤ c ≤ 1, converges in probability to ((1 − c)r − 1/2r)/(1 − 1/2r). The former quantity is of interest in the area of landscape analysis as a way to better understand problems and assess their hardness for local search heuristics. In [34], it has been shown that it may be calculated in polynomial time for any instance, and its mean value over all instances was discussed. Our results are based on a study of the variance of the number of clauses satisfied by a random assignment, and the covariance of the numbers of clauses satisfied by a random pair of assignments of an arbitrary distance. As part of this study, closed-form formulas for the expected value and vari- ance of the latter two quantities are provided. Note that all results are relevant to random r-Sat as well.",

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Berend, D & Twitto, Y 2016, The normalized autocorrelation length of random max r-Sat converges in probability to (1 − 1/2^r)/r. in D Le Berre & N Creignou (eds), Theory and Applications of Satisfiability Testing – SAT 2016 - 19th International Conference, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9710, Springer Verlag, pp. 60-76, 19th International Conference on Theory and Applications of Satisfiability Testing, SAT 2016, Bordeaux, France, 5/07/16. https://doi.org/10.1007/978-3-319-40970-2_5

The normalized autocorrelation length of random max r-Sat converges in probability to (1 − 1/2^r)/r. / Berend, Daniel; Twitto, Yochai.
Theory and Applications of Satisfiability Testing – SAT 2016 - 19th International Conference, Proceedings. ed. / Daniel Le Berre; Nadia Creignou. Springer Verlag, 2016. p. 60-76 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9710).

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

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Berend D, Twitto Y. The normalized autocorrelation length of random max r-Sat converges in probability to (1 − 1/2^r)/r. In Le Berre D, Creignou N, editors, Theory and Applications of Satisfiability Testing – SAT 2016 - 19th International Conference, Proceedings. Springer Verlag. 2016. p. 60-76. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-319-40970-2_5