Characterising the complexity of constraint satisfaction problems defined by 2-constraint forbidden patterns (original) (raw)
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The tractability of CSP classes defined by forbidden patterns
Journal of Artificial Intelligence Research, 2012
The constraint satisfaction problem (CSP) is a general problem central to computer science and artificial intelligence. Although the CSP is NP-hard in general, considerable effort has been spent on identifying tractable subclasses. The main two approaches consider structural properties (restrictions on the hypergraph of constraint scopes) and relational properties (restrictions on the language of constraint relations). Recently, some authors have considered hybrid properties that restrict the constraint hypergraph and the relations simultaneously. Our key contribution is the novel concept of a CSP pattern and classes of problems defined by forbidden patterns (which can be viewed as forbidding generic subproblems). We describe the theoretical framework which can be used to reason about classes of problems defined by forbidden patterns. We show that this framework generalises relational properties and allows us to capture known hybrid tractable classes. Although we are not close to obtaining a dichotomy concerning the tractability of general forbidden patterns, we are able to make some progress in a special case: classes of problems that arise when we can only forbid binary negative patterns (generic sub-problems in which only inconsistent tuples are specified). In this case we are able to characterise very large classes of tractable and NP-hard forbidden patterns. This leaves the complexity of just one case unresolved and we conjecture that this last case is tractable.
The Power of Arc Consistency for CSPs Defined by Partially-Ordered Forbidden Patterns
Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, 2016
Characterising tractable fragments of the constraint satisfaction problem (CSP) is an important challenge in theoretical computer science and artificial intelligence. Forbidding patterns (generic subinstances) provides a means of defining CSP fragments which are neither exclusively language-based nor exclusively structure-based. It is known that the class of binary CSP instances in which the broken-triangle pattern (BTP) does not occur, a class which includes all tree-structured instances, are decided by arc consistency (AC), a ubiquitous reduction operation in constraint solvers. We provide a characterisation of simple partially-ordered forbidden patterns which have this AC-solvability property. It turns out that BTP is just one of five such AC-solvable patterns. The four other patterns allow us to exhibit new tractable classes.
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.
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.
Full Constraint Satisfaction Problems
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.
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.
Complexity of clausal constraints over chains
Theory of Computing …, 2008
We investigate the complexity of the satisfiability problem of constraints over finite totally ordered domains. In our context, a clausal constraint is a disjunction of inequalities of the form x ≥ d and x ≤ d. We classify the complexity of constraints based on clausal patterns. A pattern abstracts away from variables and contains only information about the domain elements and the type of inequalities occurring in a constraint. Every finite set of patterns gives rise to a (clausal) constraint satisfaction problem in which all constraints in instances must have an allowed pattern. We prove that every such problem is either polynomially decidable or NPcomplete, and give a polynomial-time algorithm for recognizing the tractable cases. Some of these tractable cases are new and have not been previously identified in the literature. * The work has been supported byÉGIDE 06606ZF andÖAD Amadeus 18/2004. †
06401 Executive Summary--Complexity of Constraints}
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.
2011
Many fundamental problems in artificial intelligence, knowledge representation, and verification involve reasoning about sets and relations between sets and can be modeled as set constraint satisfaction problems (set CSPs). Such problems are frequently intractable, but there are several important set CSPs that are known to be polynomial-time tractable. We introduce a large class of set CSPs that can be solved in quadratic time. Our class, which we call E I, contains all previously known tractable set CSPs, but also some new ones that are of crucial importance for example in description logics. The class of E I set constraints has an elegant universal-algebraic characterization, which we use to show that every set constraint language that properly contains all E I set constraints already has a finite sublanguage with an NP-hard constraint satisfaction problem.
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.