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 ` literals, we obtain the problem Max-`SAT-CC. Sviridenko (Algorithmica, 30(3):398–405, 2001) designed a (1− e−1)-approximation algorithm for Max-SAT-CC. This result is tight unless P = NP (Feige, J. ACM, 45(4):634–652, 1998). Sviridenko asked if it is possible to achieve a better approximation ratio in the case of Max-`SAT-CC. We answer this question in the affirmative by presenting a randomized approximation algorithm whose approximation ratio is 1−(1− 1` ) `−ε. 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.