The Optimization Complexity of Constraint Satisfaction Problems

In 1978, Schaefer [12] considered a subclass of languages in NP and proved a “dichotomy theorem” for this class. The subclass considered were problems expressible as “constraint satisfaction problems”, and the “dichotomy theorem” showed that every language in this class is either in P, or is NP-hard. This result is in sharp contrast to a result of Ladner [9], which shows that such a dichotomy does not hold for NP, unless NP=P. We consider optimization version of the dichotomy question and show an analog of Schaefer’s result for this case. More specifically, we consider optimization version of “constraint satisfaction problems” and show that every optimization problem in this class is either solvable exactly in P, or is MAX SNP-hard, and hence not approximable to within some constant factor in polynomial time, unless NP=P. This result does not follow directly from Schaefer’s result. In particular, the set of problems that turn out to be hard in this case, is quite different from the set of languages which are shown hard by Schaefer’s result. A similar result has been independently shown by Creignou [4] using quite different techniques. [email protected]. Department of Computer Science, Stanford University, Stanford, CA 94305. Supported by a Schlumberger Foundation Fellowship, an OTL grant, and NSF Grant CCR-9357849. [email protected]. IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598.