Adding cardinality constraints to integer programs with applications to maximum satisfiability

• Published in: Information processing letters Vol. 105; no. 5; pp. 194 - 198
• Main Authors: Bläser, Markus, Heynen, Thomas, Manthey, Bodo
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 29.02.2008; Elsevier Science; Elsevier Sequoia S.A
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Approximation; Approximation algorithms; Calculus of variations and optimal control; Cardinality constraints; Computer science; control theory; systems; Exact sciences and technology; Integer programming; Mathematical analysis; Mathematics; Miscellaneous; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in mathematical programming, optimization and calculus of variations; Random variables; Randomized algorithms; Ratios; Satisfiability; Sciences and techniques of general use; Studies; Theoretical computing; Approximation algorithms; Randomized algorithms; Cardinality constraints; Satisfiability; Approximation; Computer theory; Constraint; Optimization method; Probability; Probability distribution; Approximation algorithm; Optimization; Application program; Probability space; Assignment; Design; Integer; Maximum; Information processing; Satisfiability problem; Application; Random variable; Algorithm analysis
• ISSN: 0020-0190; 1872-6119
• DOI: 10.1016/j.ipl.2007.08.024

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 ℓ 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 − 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.