Complexity and Approximability of Quantified and Stochastic Constraint Satisfaction Problems

Abstract

Let D be an arbitrary (not necessarily finite) nonempty set, let C be a finite set of constant symbols denoting arbitrary elements of D, and let S be an arbitrary finite set of finite-arity relations on D. We denote the problem of determining the satisfiability of finite conjunctions of relations in S applied to variables (to variables and symbols in C) by SAT(S) (by SAT{sub c}(S)). Here, we study simultaneously the complexity of and the existence of efficient approximation algorithms for a number of variants of the problems SAT(S) and SAT{sub c}(S), and for many different D, C, and S. These problem variants include decision and optimization problems, for formulas, quantified formulas stochastically-quantified formulas. We denote these problems by Q-SAT(S), MAX-Q-SAT(S), S-SAT(S), MAX-S-SAT(S) MAX-NSF-Q-SAT(S) and MAX-NSF-S-SAT(S). The main contribution is the development of a unified predictive theory for characterizing the the complexity of these problems. Our unified approach is based on the following basic two basic concepts: (i) strongly-local replacements/reductions and (ii) relational/algebraic representability. Let k {ge} 2. Let S be a finite set of finite-arity relations on {Sigma}{sub k} with the following condition on S: All finite arity relations on {Sigma}{sub k} can be represented as finite existentially-quantified conjunctionsmore » of relations in S applied to variables (to variables and constant symbols in C), Then we prove the following new results: (1) The problems SAT(S) and SAT{sub c}(S) are both NQL-complete and {le}{sub logn}{sup bw}-complete for NP. (2) The problems Q-SAT(S), Q-SAT{sub c}(S), are PSPACE-complete. Letting k = 2, the problem S-SAT(S) and S-SAT{sub c}(S) are PSPACE-complete. (3) {exists} {epsilon} > 0 for which approximating the problems MAX-Q-SAT(S) within {epsilon} times optimum is PSPACE-hard. Letting k =: 2, {exists} {epsilon} > 0 for which approximating the problems MAX-S-SAT(S) within {epsilon} times optimum is PSPACE-hard. (4) {forall} {epsilon} > 0 the problems MAX-NSF-Q-SAT(S) and MAX-NSF-S-SAT(S), are PSPACE-hard to approximate within a factor of n{sup {epsilon}} times optimum. These results significantly extend the earlier results by (i) Papadimitriou [Pa851] on complexity of stochastic satisfiability, (ii) Condon, Feigenbaum, Lund and Shor [CF+93, CF+94] by identifying natural classes of PSPACE-hard optimization problems with provably PSPACE-hard {epsilon}-approximation problems. Moreover, most of our results hold not just for Boolean relations: most previous results were done only in the context of Boolean domains. The results also constitute as a significant step towards obtaining a dichotomy theorems for the problems MAX-S-SAT(S) and MAX-Q-SAT(S): a research area of recent interest [CF+93, CF+94, Cr95, KSW97, LMP99].« less