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COMPLEXITY&APPROXIMABILITY OF QUANTIFIED&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. We present 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 represent ability. Some of the results extend the earlier results in [Pa85,LMP99,CF+93,CF+94O]u r techniques and results reported here also provide significant steps towards obtaining dichotomy theorems, for a number of the problems above, including the problems MAX-&-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
Schaefer's theorem for graphs
Schaefer's theorem is a complexity classification result for so-called
Boolean constraint satisfaction problems: it states that every Boolean
constraint satisfaction problem is either contained in one out of six classes
and can be solved in polynomial time, or is NP-complete.
We present an analog of this dichotomy result for the propositional logic of
graphs instead of Boolean logic. In this generalization of Schaefer's result,
the input consists of a set W of variables and a conjunction \Phi\ of
statements ("constraints") about these variables in the language of graphs,
where each statement is taken from a fixed finite set \Psi\ of allowed
quantifier-free first-order formulas; the question is whether \Phi\ is
satisfiable in a graph.
We prove that either \Psi\ is contained in one out of 17 classes of graph
formulas and the corresponding problem can be solved in polynomial time, or the
problem is NP-complete. This is achieved by a universal-algebraic approach,
which in turn allows us to use structural Ramsey theory. To apply the
universal-algebraic approach, we formulate the computational problems under
consideration as constraint satisfaction problems (CSPs) whose templates are
first-order definable in the countably infinite random graph. Our method to
classify the computational complexity of those CSPs is based on a
Ramsey-theoretic analysis of functions acting on the random graph, and we
develop general tools suitable for such an analysis which are of independent
mathematical interest.Comment: 54 page
Reasoning about temporal relations : the maximal tractable subalgebras of Allen's interval algebra
Allen's interval algebra is one of the best established formalisms for temporal reasoning. This article provides the final step in the classification of complexity for satisfiability problems over constraints expressed in this algebra. When the constraints are chosen from the full Allen's algebra, this form of satisfiability problem is known to be NP-complete. However, eighteen tractable subalgebras have previously been identified; we show here that these subalgebras include all possible tractable subsets of Allen's algebra. In other words, we show that this algebra contains exactly eighteen maximal tractable subalgebras, and reasoning in any fragment not entirely contained in one of these subalgebras is NP-complete. We obtain this dichotomy result by giving a new uniform description of the known maximal tractable subalgebras, and then systematically using a general algebraic technique for identifying maximal subalgebras with a given property