A New Lower Bound of Critical Function for (k, s)-SAT

(k,s)–SAT is the propositional satisfiable problem restricted to the instance where each clause has exactly k distinct literals and every variable occurs at most s times. It is known that there exits a critical function f such that for s≤ f(k), all (k,s)–SAT instances are satisfiable, but (k,f(k)+1)–SAT is already NP–complete(k≥ 3). It’s open whether f is computable. In this paper, analogous to the randomized algorithm for finding a two-coloring for given uniform k–hypergraph, the similar one for outputting an assignment for a given formula is presented. Based on it and the probabilistic method, we prove, for every integer k≥ 2, each formula F in (k, *)–CNF with less than 0.58 $\times \sqrt{\frac{k}{{\rm ln} k}}2^k$ clauses is satisfiable. In addition, by the Lovasz Local Lemma, we improve the previous result about the lower bound of f(k).