Satisfiability Threshold for Power Law Random 2-SAT in Configuration Model
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The Random Satisfiability problem has been intensively studied for decades. For a number of reasons the focus of this study has mostly been on the model, in which instances are sampled uniformly at random from a set of formulas satisfying some clear conditions, such as fixed density or the probability of a clause to occur. However, some non-uniform distributions are also of considerable interest. In this paper we consider Random 2-SAT problems, in which instances are sampled from a wide range of non-uniform distributions. The model of random SAT we choose is the so-called configuration model, given by a distribution ξ for the degree (or the number of occurrences) of each variable. Then to generate a formula the degree of each variable is sampled from ξ, generating several clones of the variable. Then 2-clauses are created by choosing a random paritioning into 2-element sets on the set of clones and assigning the polarity of literals at random. Here we consider the random 2-SAT problem in the configuration model for power-law-like distributions ξ. More precisely, we assume that ξ is such that its right tail F_ξ(x) satisfies the conditions Wℓ^-α< F_ξ(ℓ)< Vℓ^-α for some constants V,W. The main goal is to study the satisfiability threshold phenomenon depending on the parameters α,V,W. We show that a satisfiability threshold exists and is determined by a simple relation between the first and second moments of ξ.
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