Complexity and Approximability of Quantified and Stochastic Constraint Satisfaction Problems

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 and T 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 SATc(S).) Here, we study simultaneously the complexity of decision, counting, maximization and approximate maximization problems, for unquantified, quantified and stochastically quantified formulas. sent simple yet general techniques to characterize simultaneously, the complexity or efficient approximability of a number of versions/variants of the problems SAT(S), Q-SAT(S), S-SAT(S),MAX-Q-SAT(S) etc., for many different such D,C,S,T. These versions/variants include decision, counting, maximization and approximate maximization problems, for unquantified, quantified and stochastically quantified formulas. Our unified approach is based on the following two basic concepts: (i) strongly-local replacements/reductions and (ii) relational/algebraic representability. f the results extend the earlier results in [Pa85,LMP99,CF+93,CF+94] Our techniques and results reported here also provide significant steps towards obtaining dichotomy theorems, for a number of the problems above, including the problems MAX-Q-SAT(S), and MAX-S-SAT(S). The discovery of such dichotomy theorems, for unquantified formulas, has received significant recent attention in the literature [CF+93, CF+94, Cr95, KSW97].