The Complexity of Equality Constraint Languages

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, 2006

Feder and Vardi have conjectured that all constraint satisfaction problems to a fixed structure (constraint language) are polynomial or NP-complete. This so-called Dichotomy Conjecture remains open, although it has been proved in a number of special cases. Most recently, Bulatov has verified the conjecture for conservative structures, i.e., structures which contain all possible unary relations.

Theoretical Computer Science

An equality language is a relational structure with infinite domain whose relations are first-order definable in equality. We classify the extensions of the quantified constraint satisfaction problem over equality languages in which the native existential and universal quantifiers are augmented by some subset of counting quantifiers. In doing this, we find ourselves in various worlds in which dichotomies or trichotomies subsist.

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.

Automated Reasoning, 2004

Given a finite set of vectors over a finite totally ordered domain, we study the problem of computing a constraint in conjunctive normal form such that the set of solutions for the produced constraint is identical to the original set. We develop an efficient polynomial-time algorithm for the general case, followed by specific polynomial-time algorithms producing Horn, dual Horn, and bijunctive constraints for sets of vectors closed under the operations of conjunction, disjunction, and median, respectively. We also consider the affine constraints, analyzing them by means of computer algebra. Our results generalize the work of Dechter and Pearl on relational data, as well as the papers by Hébrard and Zanuttini. They also complete the results of Hähnle et al. on multivalued logics and Jeavons et al. on the algebraic approach to constraints. We view our work as a step toward a complete complexity classification of constraint satisfaction problems over finite domains.

2006

In a constraint satisfaction problem, the goal is to find an assignment of values to a given set of variables so that certain specified constraints are satisfied. Constraint satisfaction problems were introduced in the 1970s to model computational problems encountered in scene processing. It was quickly realized, however, that constraint satisfaction gives rise to a powerful general framework in which a wide variety of combinatorial problems can be expressed. As a matter of fact, it has been asserted that "Constraint satisfaction has a unitary theoretical model with myriad practical applications" (A. Mackworth, Foreword to Constraint Processing by Rina Dechter, 2003). Indeed, in artificial intelligence and other areas of computer science the constraint satisfaction framework is widely regarded as a versatile and efficient way of modeling and solving a number of real-world problems, such as planning and scheduling, frequency assignment problems, image processing, programming language analysis, and natural language understanding. In database theory, it has been shown that the key problem of conjunctive-query evaluation is actually equivalent to the constraint satisfaction problem. Furthermore, certain central problems in combinatorial optimization can be represented as constraint satisfaction problems. Thus, nowadays constraint satisfaction problems (CSPs) are ubiquitous in many different areas of computer science, from artificial intelligence and database systems to circuit design, network optimization, and theory of programming languages. Consequently, it is important to analyze and pinpoint the computational complexity of certain algorithmic tasks related to constraint satisfaction. These include determining if a CSP has a solution (and, if so, finding such a solution), counting the number of solutions of a CSP, enumerating all solutions of a CSP, and finding the biggest number of constraints that can be simultaneously satisfied, if a CSP is unsatisfiable. Complexity-theoretic results about these tasks may have direct impact on, for instance, the design and processing of database query languages, or strategies in data-mining, or the design and implementation of planners. During the past two decades, an impressive array of diverse techniques from mathematical fields, such as propositional logic, model theory, Boolean function theory, universal algebra and combinatorics, have been used to analyze the computational complexity of algorithmic tasks related to CSPs. Although significant progress has been made on several fronts, some of the central questions remain unsolved so far; perhaps the most prominent of these is to obtain a complete classification of the complexity of CSPs over an arbitrary, but fixed, finite domain. One of the main aims of the Dagstuhl Seminar on Complexity of Constraints was to bring together researchers from all areas of activity in constraint satisfaction, so that they can communicate state-of-theart advances and embark on a systematic interaction that will enhance the synergy between the different areas.

Lecture Notes in Computer Science, 2004

We consider the Boolean constraint isomorphism problem, that is, the problem of determining whether two sets of Boolean constraint applications can be made equivalent by renaming the variables. We show that depending on the set of allowed constraints, the problem is either coNP-hard and GI-hard, equivalent to graph isomorphism, or polynomial-time solvable. This establishes a complete classification of the complexity of the problem, and moreover, it identifies exactly all those cases in which Boolean constraint isomorphism is polynomial-time manyone equivalent to graph isomorphism, the best-known and best-examined isomorphism problem in theoretical computer science.

Lecture Notes in Computer Science, 2009

Constraint Satisfaction Problems (CSP) constitute a convenient way to capture many combinatorial problems. The general CSP is known to be NPcomplete, but its complexity depends on a parameter, usually a set of relations, upon which they are constructed. Following the parameter, there exist tractable and intractable instances of CSPs. In this paper we show a dichotomy theorem for every finite domain of CSP including also disjunctions. This dichotomy condition is based on a simple condition, allowing us to classify monotone CSPs as tractable or NP-complete. We also prove that the meta-problem, verifying the tractability condition for monotone constraint satisfaction problems, is fixed-parameter tractable. Moreover, we present a polynomial-time algorithm to answer this question for monotone CSPs over ternary domains. J = {(x, y, z, w) ∈ D 4 | (x = y) ∨ (z = w)} in a classical constraint satisfaction problem reduces it to a monotone constraint satisfaction problem.

CoLogNet Publications, 2002

Many constraint satisfaction problems (csp's) are formulated with 0/1 variables. Sometimes this is a natural encoding, sometimes it is as a result of a reformulation of the problem, other times 0/1 variables make up only a part of the problem. Frequently we have constraints that restrict the sum of the values of variables. This can be encoded as a simple summation of the variables. However, since variables can only take 0/1 values we can also use an occurrence constraint, e.g. the number of occurrences of 1 must be k. Would this make a difference? Similarly, problems may use channelling constraints and encode these as a biconditional such as P ↔ Q (i.e. P if and only if Q). This can also be encoded in a number of ways. Might this make a difference as well? We attempt to answer these questions, using a variety of problems and two constraint programming toolkits. We show that even minor changes to the formulation of a constraint can have a profound effect on the run time of a constraint program and that these effects are not consistent across constraint programming toolkits. This leads us to a cautionary note for constraint programmers: take note of how you encode constraints, and don't assume computational behaviour is toolkit independent.

2005

We give a clear picture of the tractability/intractability frontier for quantified constraint satisfaction problems (QCSPs) under structural restrictions. On the negative side, we prove that checking QCSP satisfiability remains PSPACE-hard for all known structural properties more general than bounded treewidth and for the incomparable hypergraph acyclicity. Moreover, if the domain is not fixed, the problem is PSPACE-hard even for tree-shaped constraint scopes. On the positive side, we identify relevant tractable classes, including QCSPs with prefix ∃∀ having bounded hypertree width, and QCSPs with a bounded number of guards. The latter are solvable in polynomial time without any bound on domains or quantifier alternations.