Note on strong refutation algorithms for random k -SAT formulas Hiˆe . p H`an 1 , Yury Person 2 , and Mathias Schacht Institut f¨ ur Informatik Humboldt-Universit¨at zu Berlin Unter den Linden 6, D-10099 Berlin, Germany Abstract We present a simple strong refutation algorithm for random k-SAT formulas. Our al- gorithm applies to random k-SAT formulas on n variables with ω(n)n (k+1)/2 clauses for any ω(n) →∞. In contrast to the earlier results of Coja-Oghlan, Goerdt, and Lanka (for k =3, 4) and Coja-Oghlan, Cooper, and Frieze (for k ≥ 5), which address the same problem for even sparser formulas our algorithm is more elementary. 1 Introduction The k-SAT problem is among the best studied NP-complete problems. We consider strong refutation algorithms for random k-SAT. Let X n = {x 1 ,...,x n } be a set of n propositional variables, let p = p(n) ∈ [0, 1], and let F k (n, p) be the probability space over all k-SAT formulas on X n , for which each of the (2n) k possible (ordered) k-clauses will be included independently with probability p. It is well-known that for p  n 1-k with high probability a random formula F ∈F k (n, p) is not satisfiable. However, there are no efficient refutation algorithms known. We are interested in deterministic algorithms which w.h.p. reject a k-SAT formula from F k (n, p) for p  n 1-k , but which never reject a satisfiable formula. An algorithm is a strong refutation algorithm if w.h.p. for F ∈F k (n, p) it approximates unsat(F ) by a factor of (1 - ε) and never outputs a number 1 Author is supported by DFG within the RTG “Methods for Discrete Structures”. 2 Author is supported by GIF grant no. I-889-182.6/2005