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The Normalized Autocorrelation Length of Random Max \(r\)-Sat Converges in Probability to \((1-1/2^r)/r\)

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Theory and Applications of Satisfiability Testing – SAT 2016 (SAT 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9710))

Abstract

In this paper we show that the so-called normalized autocorrelation 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 \(d=cn\), \(0 \le c \le 1\), 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 variance of the latter two quantities are provided. Note that all results are relevant to random \(r\)-Sat as well.

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Acknowledgment

The work of the second author was partially supported by the Lynne and William Frankel Center for Computer Science.

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  1. Ben-Gurion University, Beer Sheva, 84105, Israel

    Daniel Berend & Yochai Twitto

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  1. Daniel Berend
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  2. Yochai Twitto
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Correspondence to Yochai Twitto .

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  1. Aix-Marseille Université, Marseille, France

    Nadia Creignou

  2. Université d'Artois, Lens, France

    Daniel Le Berre

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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: Creignou, N., Le Berre, D. (eds) Theory and Applications of Satisfiability Testing – SAT 2016. SAT 2016. Lecture Notes in Computer Science(), vol 9710. Springer, Cham. https://doi.org/10.1007/978-3-319-40970-2_5

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  • DOI: https://doi.org/10.1007/978-3-319-40970-2_5

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