Adding cardinality constraints to integer programs with applications to maximum satisfiability

Max-SAT-CC is the following optimization problem: Given a formula in CNF and a bound k, find an assignment with at most k variables being set to true that maximizes the number of satisfied clauses among all such assignments. If each clause is restricted to have at most e literals, we obtain the problem Max-lSAT-CC. Sviridenko [Algorithmica 30 (3) (2001) 398-405] designed a (1-e(-1))-approximation algorithm for Max-SAT-CC. This result is tight unless P = NP [U. Feige, J. ACM 45 (4) (1998) 634-652]. Sviridenko asked if it is possible to achieve a better approximation ratio in the case of Max-lSAT-CC. We answer this question in the affirmative by presenting a randomized approximation algorithm whose approximation ratio is 1-(1-1/l)(l)-epsilon. To do this, we develop a general technique for adding a cardinality constraint to certain integer programs. Our algorithm can be derandomized using pairwise independent random variables with small probability space. (c) 2007 Elsevier B.V. All rights reserved.