The Complexity of Quantified Constraint Satisfaction: Collapsibility, Sink Algebras, and the Three-Element Case

Abstract

The constraint satisfaction probem (CSP) is a well-acknowledged framework in which many combinatorial search problems can be naturally formulated. The CSP may be viewed as the problem of deciding the truth of a logical sentence consisting of a conjunction of constraints, in front of which all variables are existentially quantified. The quantified constraint satisfaction problem (QCSP) is the generalization of the CSP where universal quantification is permitted in addition to existential quantification. The general intractability of these problems has motivated research studying the complexity of these problems under a restricted constraint language, which is a set of relations that can be used to express constraints.This paper introduces collapsibility, a technique for deriving positive complexity results on the QCSP. In particular, this technique allows one to show that, for a particular constraint language, the QCSP reduces to the CSP. We show that collapsibility applies to three known tractable cases of the QCSP that were originally studied using disparate proof techniques in different decades: QUANTIFIED 2-SAT (Aspvall, Plass, and Tarjan 1979), QUANTIFIED HORN-SAT (Karpinski, Kleine Büning, and Schmitt 1987), and QUANTIFIED AFFINE-SAT (Creignou, Khanna, and Sudan 2001). This reconciles and reveals common structure among these cases, which are describable by constraint languages over a two-element domain. In addition to unifying these known tractable cases, we study constraint languages over domains of larger size.

The Complexity of Valued Constraint Satisfaction Problems

“…Creignou et al have studied quantified CSPs over Boolean domains [87]. More work on quantified CSPs was initiated by [32] and Chen's thesis [53]; see also [55]. A complexity classification has been obtained for quantified equality constraint languages [23] and relatively quantified CSPs [24,110] (that is, quantified CSPs with all unary constraints; in the case of non-quantified CSPs, these are known as conservative CSPs).…”

Decomposing Quantified Conjunctive (or Disjunctive) Formulas

“…A fragment of first-order logic under current scrutiny [10,20,11,12,13,14,15,21] is the class of quantified conjunctive sentences, which is the generalization of primitive positive sentences where both quantifiers are admitted, that is, sentences of the form Q 1 v 1 . .…”

A Tetrachotomy for Positive First-Order Logic without Equality

“…The most successful approach to date, and the method used in [4], has been the so-called algebraic method, in which the problem of classification reverts to classes of functions under which the relevant relational structures are invariant. A similar algebraic approach has been successful in the study of the evaluation problem for positive Horn sentences, QCSP(D), and, while no formal trichotomy has there been conjectured, the only known attainable complexities are P, NP-complete and Pspace-complete [5], [6].…”