The Optimization Complexity of Structure Maximization Problems

In 1978, Schaefer [18] considered a subclass of languages in NP. This class, which generalizes the satisfiability problem, has as instances general constraints (of finite arity) placed on a collection of n boolean variables and the aim is to determine if there exists an assignment to the variables satisfying all constraints. Schaefer’s result classifies the hardness of deciding such problems based on the nature of the allowed constraints and shows that for every family of constraints, the decision problem is either in P or is NP-hard. We consider an optimization problem built on Schaefer’s problem: the constraints are boolean constraints as in Schaefer’s case, but the objective now is to find a feasible solution which maximimizes the number of ones (more generally, we maximize the weight of a solution given a nonnegative weight function on the variables). We determine the approximability of every problem in this class of problems and shows this lies in one of five classes: exactly optimizable in poly-time, approximable to within constant factors in poly-time, approximable to within polynomial factors in poly-time, not approximable but decidable, and not decidable (unless NP=P). Our work builds on the impressive collection of non-approximability results derived over the last five years, initiated by the work of Feige, Goldwasser, Lovasz, Safra and Szegedy. However much of the past work has been enumerative in its approach, listing out specific problems and then proving hardness results for such problems. Our approach adds to this body of work by extracting exhaustive results from these – namely we build a large (infinite) collection of problems and then characterize fairly precisely the approximability of every one of these problems. Our work is closer to the study of Creignou [5] and Khanna and Sudan [11] who present similar exhaustive results for a different class of optimization problems. Their classes, however, fail to include any non-constant approximable problem. Our study thus seems to classify a richer variety of optimization problems. sanjeev@theory.stanford.edu. Department of Computer Science, Stanford University, Stanford, CA 94305. Supported by a Schlumberger Foundation Fellowship, an OTL grant, and NSF Grant CCR-9357849. ymadhu@watson.ibm.com. IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598. zdpw@watson.ibm.com. IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598.