The Complexity of Quantified Constraint Satisfaction Problems under Structural Restrictions (original) (raw)
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Low-level dichotomy for quantified constraint satisfaction problems
Information Processing Letters, 2011
Building on a result of Larose and Tesson for constraint satisfaction problems (CSPs), we uncover a dichotomy for the quantified constraint satisfaction problem QCSP(B), where B is a finite structure that is a core. Specifically, such problems are either in ALogtime or are L-hard. This involves demonstrating that if CSP(B) is first-order expressible, and B is a core, then QCSP(B) is in ALogtime. We show that the class of B such that CSP(B) is first-order expressible (indeed, trivially true) is a microcosm for all QCSPs. Specifically, for any B there exists a C such that CSP(C) is trivially true, yet QCSP(B) and QCSP(C) are equivalent under logspace reductions.
The Complexity of Problems for Quantified Constraints
Theory of Computing Systems / Mathematical Systems Theory, 2007
In this paper we will look at restricted versions of the evaluation problem, the model checking problem, the equivalence problem, and the counting problem for quantified propositional formulas, both with and without bound on the number of quantifier alternations. The restrictions are such that we consider formulas in conjunctive normal-form with restricted types of clauses (e.g., positive, Horn, linear, etc.). For each of these algorithmic goals we will obtain full complexity classifications, exhibiting on the one hand severe syntactic restrictions of the original problems that are still computationally hard, and on the other hand non-trivial subcases that admit efficient solution algorithms. Generalizing these results to non Boolean domains, we obtain a number of hardnes results for quantified constraints over arbitrary finite universes. Supported in part by the following grants: DFG Vo 630/5-1, 630/5-2,ÉGIDE 05835SH, DAAD D/0205776. Some of the results reported in Sect. 5.2 of this paper already appeared in ECCC Report 05-024.
SIAM Journal on Computing, 2008
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.
Quantified Constraints and Containment Problems
2008 23rd Annual IEEE Symposium on Logic in Computer Science, 2008
We study two containment problems related to the quantified constraint satisfaction problem (QCSP). Firstly, we give a combinatorial condition on finite structures A and B that is necessary and sufficient to render QCSP(A) ⊆ QCSP(B). The required condition is the existence of a positive integer r such that there is a surjective homomorphism from the power structure A r to B. We note that this condition is already necessary to guarantee containment of the Π 2 restriction of QCSP, that is Π 2-CSP(A) ⊆ Π 2-CSP(B). Since we are able to give an effective bound on such an r, we provide a decision procedure for the model containment problem with non-deterministic double-exponential time complexity. Secondly, we prove that the entailment problem for quantified conjunctive-positive first-order logic is decidable. That is, given two sentences ϕ and ψ of first-order logic with no instances of negation or disjunction, we give an algorithm that determines whether ϕ → ψ is true in all structures (models). Our result is in some sense tight, since we show that the entailment problem for positive firstorder logic (i.e. quantified conjunctive-positive logic plus disjunction) is undecidable.
A Characterisation of First-Order Constraint Satisfaction Problems
Logical Methods in Computer Science, 2007
We describe simple algebraic and combinatorial characterisations of finite relational core structures admitting finitely many obstructions. As a consequence, we show that it is decidable to determine whether a constraint satisfaction problem is first-order definable: we show the general problem to be NP-complete, and give a polynomial-time algorithm in the case of cores. A slight modification of this algorithm provides, for firstorder definable CSP's, a simple poly-time algorithm to produce a solution when one exists. As an application of our algebraic characterisation of first order CSP's, we describe a large family of L-complete CSP's.
Beyond NP: Arc-Consistency for Quantified Constraints
Lecture Notes in Computer Science, 2002
The generalization of the satisfiability problem with arbitrary quantifiers is a challenging problem of both theoretical and practical relevance. Being PSPACE-complete, it provides a canonical model for solving other PSPACE tasks which naturally arise in AI. Effective SAT-based solvers have been designed very recently for the special case of boolean constraints. We propose to consider the more general problem where constraints are arbitrary relations over finite domains. Adopting the viewpoint of constraint-propagation techniques so successful for CSPs, we provide a theoretical study of this problem. Our main result is to propose quantified arc-consistency as a natural extension of the classical CSP notion.
Non-dichotomies in Constraint Satisfaction Complexity
Automata, Languages and Programming, 2008
We show that every computational decision problem is polynomialtime equivalent to a constraint satisfaction problem (CSP) with an infinite template. We also construct for every decision problem L an ω-categorical template Γ such that L reduces to CSP(Γ ) and CSP(Γ ) is in coNP L (i.e., the class coNP with an oracle for L). CSPs with ω-categorical templates are of special interest, because the universal-algebraic approach can be applied to study their computational complexity. Furthermore, we prove that there are ω-categorical templates with coNP-complete CSPs and ω-categorical templates with coNP-intermediate CSPs, i.e., problems in coNP that are neither coNP-complete nor in P (unless P=coNP). To construct the coNP-intermediate CSP with ω-categorical template we modify the proof of Ladner's theorem. A similar modification allows us to also prove a non-dichotomy result for a class of left-hand side restricted CSPs, which was left open in . We finally show that if the so-called local-global conjecture for infinite constraint languages (over a finite domain) is false, then there is no dichotomy for the constraint satisfaction problem for infinite constraint languages.
HyperConsistency Width for Constraint Satisfaction: Algorithms and Complexity Results
Lecture Notes in Computer Science, 2009
Generalized hypertree width (short: ghw) is a concept that leads to a large class of efficiently solvable CSP instances, whose associated recognition problem (of checking whether the ghw of a CSP is bounded by a constant k) is however known to be NP-hard. An elegant way to circumvent this intractability has recently been proposed in the literature, by means of a "no-promise" approach solving CSPs of bounded ghw without the need of actually computing a generalized hypertree decomposition. In fact, despite the conceptual relevance of this approach, its computational issues have not yet been investigated and, indeed, precise bounds on the running time of the no-promise algorithm are missing. The first contribution of this paper is precisely to fill this gap. Indeed, the computational complexity of the no-promise approach is analyzed, by exploiting an intuitive characterization relying on the notion of hyperconsistency width. It turns out that, in the basic formulation, the approach is hardly suited for practical applications mainly because of its bad scaling in the size of the constraint database. Motivated by these news and based on a variant of hyperconsistency width, a different and more efficient method to decide whether CSPs of bounded ghw admit solutions is then provided. Importantly, the improved method exhibits the same scaling as current evaluation algorithms for instances of bounded hypertree width, nonetheless allowing to isolate a larger class of queries. Finally, to give a complete picture of the complexity issues of the no-promise approach, the problems of computing one solution and of enumerating all the solutions are also studied.
Tractability in constraint satisfaction problems: a survey
Constraints, 2015
Even though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP. 1. automatic recognition and resolution of easy instances within general-purpose solvers, supported by ANR Project ANR-10-BLAN-0210 and EPSRC grant EP/L021226/1.
Tractable Structures for Constraint Satisfaction with Truth Tables
Theory of Computing Systems, 2009
The way the graph structure of the constraints influences the complexity of constraint satisfaction problems (CSP) is well understood for bounded-arity constraints. The situation is less clear if there is no bound on the arities. In this case the answer depends also on how the constraints are represented in the input. We study this question for the truth table representation of constraints. We introduce a new hypergraph measure adaptive width and show that CSP with truth tables is polynomial-time solvable if restricted to a class of hypergraphs with bounded adaptive width. Conversely, assuming a conjecture on the complexity of binary CSP, there is no other polynomial-time solvable case.