Precise Upper and Lower Bounds for the Monotone Constraint Satisfaction Problem

Abstract

The monotone constraint satisfaction problem (MCSP) is the problem of, given an existentially quantified positive formula, decide whether this formula has a model. This problem is a natural generalization of the constraint satisfaction problem, which can be seen as the problem of determining whether a conjunctive formula has a model. In this paper we study the worst-case time complexity, measured with respect to the number of variables, n, of the MCSP problem parameterized by a constraint language $$\varGamma $$ (MCSP $$(\varGamma )$$ ). We prove that the complexity of the NP-complete MCSP $$(\varGamma )$$ problems on a given finite domain D falls into exactly $$|D| - 1$$ cases and ranges from $$O(2^{n})$$ to $$O(|D|^n)$$ . We give strong lower bounds and prove that MCSP $$(\varGamma )$$ , for any constraint language $$\varGamma $$ over any finite domain, is solvable in $$O(|D'|^n)$$ time, where $$D'$$ is the domain of the core of $$\varGamma $$ , but not solvable in $$O(|D'|^{\delta n})$$ time for any $$\delta < 1$$ , unless the strong exponential-time hypothesis fails. Hence, we obtain a complete understanding of the worst-case time complexity of MCSP $$(\varGamma )$$ for constraint languages over arbitrary finite domains.