The Satisfiability Threshold for Non-Uniform Random 2-SAT
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Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. Its worst-case hardness lies at the core of computational complexity theory, for example in the form of NP-hardness and the (Strong) Exponential Time Hypothesis. In practice however, SAT instances can often be solved efficiently. This contradicting behavior has spawned interest in the average-case analysis of SAT and has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, most theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a non-uniform distribution of the variables, which can result in distributions closer to industrial SAT instances. We study satisfiability thresholds of non-uniform random 2-SAT with n variables and m clauses and with an arbitrary probability distribution (p_i)_i∈[n] with p_1 > p_2 >...> p_n > 0 over the n variables. We show for p_1^2=Θ(∑_i=1^n p_i^2) that the asymptotic satisfiability threshold is at m=Θ( (1-∑_i=1^n p_i^2)/(p_1·(∑_i=2^n p_i^2)^1/2) ) and that it is coarse. For p_1^2=o(∑_i=1^n p_i^2) we show that there is a sharp satisfiability threshold at m=(∑_i=1^n p_i^2)^-1. This result generalizes the seminal works by Chvatal and Reed [FOCS 1992] and by Goerdt [JCSS 1996].
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