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On the maximum satisfiability of random formulas
 Proceedings of 44th Symposium on Foundations of Computer Science (FOCS 2003), IEEE Computer Society
, 2003
"... Maximum satisfiability is a canonical NPhard optimization problem that appears empirically hard for random instances. In particular, its apparent hardness on random kCNF formulas of certain densities was recently suggested by Feige as a starting point for studying inapproximability. At the same ti ..."
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Maximum satisfiability is a canonical NPhard optimization problem that appears empirically hard for random instances. In particular, its apparent hardness on random kCNF formulas of certain densities was recently suggested by Feige as a starting point for studying inapproximability. At the same time, it is rapidly becoming a canonical problem for statistical physics. In both of these realms, evaluating new ideas relies crucially on knowing the maximum number of clauses one can typically satisfy in a random kCNF formula. In this paper we give asymptotically tight estimates for this quantity. Specifically, let us say that a kCNF is psatisfiable if there exists a truth assignment satisfying 1 −2 −k +p2 −k of all clauses (observe that every kCNF is 0satisfiable). Also, let Fk(n, m) denote a random kCNF on n variables formed by selecting uniformly and independently m out of all 2 k () n possible kclauses. k Let τ(p) = 2 k ln 2/(p + (1 − p)ln(1 − p)). It is easy to prove that for every k ≥ 2 and every p ∈ (0,1], if r ≥ τ(p) then the probability that Fk(n, rn) is psatisfiable tends to 0 as n → ∞. We prove that there exists a sequence δk → 0 such that if r ≤ (1 − δk)τ(p) then the probability that Fk(n, rn) is psatisfiable tends to 1 as n → ∞. The sequence δk tends to 0 exponentially fast in k. Indeed, even for moderate values of k, e.g. k = 10, our result gives very tight bounds for the number of satisfiable clauses in a random kCNF. In particular, for k> 2 it improves upon all previously known such bounds. ∗ Part of this work was done while visiting UC Berkeley. † Research supported by NSF Grant DMS0104073 and a Miller Professorship at UC Berkeley. 1 1
On the Fraction of Satisfiable Clauses in Typical Formulas
 Extended Abstract in FOCS’03
, 2003
"... Given n Boolean variables x1,..., xn, a kclause is a disjunction of k literals, where a literal is a variable or its negation. A kCNF formula is a conjunction of a finite number of kclauses. Call such a formula psatisfiable if there exists a truth assignment satisfying a fraction 12k +p 2k of ..."
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Given n Boolean variables x1,..., xn, a kclause is a disjunction of k literals, where a literal is a variable or its negation. A kCNF formula is a conjunction of a finite number of kclauses. Call such a formula psatisfiable if there exists a truth assignment satisfying a fraction 12k +p 2k of all clauses and observe that every kCNF formula is 0satisfiable. Our goal is to determine for which ratios of clauses to variables, a typical kCNF formula is psatisfiable. Let Fk (n, m) denote a formula on n variables formed by selecting uniformly and independently m out of all (2n) k
Examples of Macroeconomic and NonEconomic Dynamic Models That are Not Self Averaging
, 2010
"... This paper describes examples of nonself averaging phenomena drawn from macroeconomic and physics fields. They are models of random clusters, such as PoissonDirichlet models, urn models, and models of random transport through disordered media. In particular, we discuss several threeparameter exte ..."
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This paper describes examples of nonself averaging phenomena drawn from macroeconomic and physics fields. They are models of random clusters, such as PoissonDirichlet models, urn models, and models of random transport through disordered media. In particular, we discuss several threeparameter extension of the two parameter PoissonDirichlet model. These three parameter models inherit nonself averaging asymptotic behavior of the twoparameter PD(α, θ) model, with positive α. Models of random additive types and random multiplicative types are mentioned. A sufficient condition for models to be nonself averaging is also presented. 1