Bounds on the satisfiability threshold for power law distributed random SAT

Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. The worst-case hardness of SAT lies at the core of computational complexity theory. The averagecase analysis of SAT has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, nearly all 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 scale-free distribution of the variables, which results in distributions closer to industrial SAT instances. We study random k-SAT on n variables, m = ϵ(n) clauses, and a power law distribution on the variable occurrences with exponent β. We observe a satisfiability threshold at β≤(2k-1)/(k-1). This threshold is tight in the sense that instances with β ≥ (2k-1)/(k-1)-ϵ for any constant ϵ > 0 are unsatisfiable with high probability (w. h. p.). For β > (2k-1)/(k-1)+ ϵ, the picture is reminiscent of the uniform case: instances are satisfiable w. h. p. for sufficiently small constant clause-variable ratios m/n; they are unsatisfiable above a ratio m/n that depends on β

Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT

Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. The worst-case hardness of SAT lies at the core of computational complexity theory. The average-case analysis of SAT has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, nearly all 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 scale-free distribution of the variables, which results in distributions closer to industrial SAT instances. We study random k-SAT on n variables, m = Theta(n) clauses, and a power law distribution on the variable occurrences with exponent beta. We observe a satisfiability threshold at beta = (2k-1)/(k-1). This threshold is tight in the sense that instances with beta 0 are unsatisfiable with high probability (w.h.p.). For beta >= (2k-1)/(k-1)+epsilon, the picture is reminiscent of the uniform case: instances are satisfiable w.h.p. for sufficiently small constant clause-variable ratios m/n; they are unsatisfiable above a ratio m/n that depends on beta

The Satisfiability Threshold for Non-Uniform Random 2-SAT

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 in[n]} with p_1 >=slant p_2 >=slant ... >=slant p_n>0 over the n variables. We show for p_{1}^2=Theta (sum_{i=1}^n p_i^2) that the asymptotic satisfiability threshold is at {m=Theta ((1-{sum_{i=1}^n p_i^2})/(p_1 * (sum_{i=2}^n p_i^2)^{1/2}))} and that it is coarse. For p_{1}^2=o (sum_{i=1}^n p_i^2) we show that there is a sharp satisfiability threshold at m=(sum_{i=1}^n p_i^2)^{-1}. This result generalizes the seminal works by Chvatal and Reed [FOCS 1992] and by Goerdt [JCSS 1996]

Scale-Free Random SAT Instances

We focus on the random generation of SAT instances that have properties similar to real-world instances. It is known that many industrial instances, even with a great number of variables, can be solved by a clever solver in a reasonable amount of time. This is not possible, in general, with classical randomly generated instances. We provide a different generation model of SAT instances, called \emph{scale-free random SAT instances}. It is based on the use of a non-uniform probability distribution $P(i)\sim i^{-\beta}$ to select variable $i$, where $\beta$ is a parameter of the model. This results into formulas where the number of occurrences $k$ of variables follows a power-law distribution $P(k)\sim k^{-\delta}$ where $\delta = 1 + 1/\beta$. This property has been observed in most real-world SAT instances. For $\beta=0$, our model extends classical random SAT instances. We prove the existence of a SAT-UNSAT phase transition phenomenon for scale-free random 2-SAT instances with $\beta<1/2$ when the clause/variable ratio is $m/n=\frac{1-2\beta}{(1-\beta)^2}$. We also prove that scale-free random k-SAT instances are unsatisfiable with high probability when the number of clauses exceeds $\omega(n^{(1-\beta)k})$. %This implies that the SAT/UNSAT phase transition phenomena vanishes when $\beta>1-1/k$, and formulas are unsatisfiable due to a small core of clauses. The proof of this result suggests that, when $\beta>1-1/k$, the unsatisfiability of most formulas may be due to small cores of clauses. Finally, we show how this model will allow us to generate random instances similar to industrial instances, of interest for testing purposes