Satisfiability Decay along Conjunctions of Pseudo-Random Clauses

Abstract. k-SAT is a fundamental constraint satisfaction problem. It involves S(m), the satisfaction set of the conjunction of m clauses, each clause a disjunction of k literals. The problem has many theoretical, algorithmic and practical aspects. When the clauses are chosen at random it is anticipated (but not fully proven) that, as the density parameter m/n (n the number of variables) grows, the transition of S(m) to being empty, is abrupt: It has a "sharp threshold", with probability 1 - o(1). In this article we replace the random ensemble analysis by a pseudo-random one: Derive the decay rule for individual sequences of clauses, subject to combinatorial conditions, which in turn hold with probability 1 - o(1). This is carried out under the big relaxation that k is not constant but k = log n, or even r log log n. Then the decay of S is slow, "near-perfect" (like a radioactive decay), which entails sharp thresholds for the transition-time of S below any given lev...

• Type: Journal
• Year: 2006
• Where: IGPL
• Authors: Eli Shamir
• Keywords: Fundamental Constraint Satisfaction, IGPL 2006, Practical Aspects, Sharp Thresholds