Probabilistic characterization of random Max r-Sat

In this paper we provide a probabilistic characterization of the random Max r-Sat problem. We study 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. Closed-form formulas for the expected value and the variance of these quantities are provided. We asymptotically and probabilistically analyze these formulas and use them to gain insights on the similarity of instances. Based on the above probabilistic characterization, we 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). Our main result is that the so-called normalized autocorrelation length of Max r-Sat converges in probability to (1−1∕2^r)∕r. The latter 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. A former result regarding the same quantity only expressed it in terms of Walsh coefficients. All our results apply to random r-Sat as well.