Journal of Information and Computing Science, JIC, Vol1 2006 No1

( , ) k s SAT − is the propositional satisfiable problem restricted to instances where each clause has exactly k distinct literals and every variable occurs at most s times. It is known that there exits an exponential function f such that for ( ) s f k ≤ , all ( , ) k s SAT − instances are satisfiable, but ( , ( ) 1) k f k SAT + − is already NPcomplete ( 3 k ≥ ). Therefore, we call the function (.) f critical function. Exact values of (.) f are only known for 3 k = and 4 k = , and it’s open whether (.) f is computable. The best known lower and upper bounds on ( ) f k are (2 / ) k k Ω and (2 / ) k k Ο , where 3 log 1 0.26 α = − ≈ , respectively. In this paper, analogous to the randomized algorithm for finding two coloring of k − uniform hypergraph, we first present a similar randomize algorithm for outputting an assignment for a given formula. Based on it and by probabilistic method, we prove that, for every integer 2 k ≥ , each formula F in ( ,*) k CNF − with at most 0.58 2 ln k k k × clauses is satisfiable. In addition, by the Lovász Local lemma, we get a new lower bound of ( ) f k , ( 2 / ) ln k k k k Ω , which improves the result (2 / ) k k Ω .