Constraint Satisfaction Problems over the Integers with Successor (original) (raw)

Distance constraint satisfaction problems

Mathematical Foundations of Computer Science 2010, 2010

We study the complexity of constraint satisfaction problems for templates Γ that are first-order definable in (Z; succ), the integers with the successor relation. Assuming a widely believed conjecture from finite domain constraint satisfaction (we require the tractability conjecture by Bulatov, Jeavons and Krokhin in the special case of transitive finite templates), we provide a full classification for the case that Γ is locally finite (i.e., the Gaifman graph of Γ has finite degree). We show that one of the following is true: The structure Γ is homomorphically equivalent to a structure with a certain majority polymorphism (which we call modular median) and CSP(Γ) can be solved in polynomial time, or Γ is homomorphically equivalent to a finite transitive structure, or CSP(Γ) is NP-complete.

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.

The complexity of recursive constraint satisfaction problems

Annals of Pure and Applied Logic, 2009

We investigate the complexity of finding solutions to infinite recursive constraint satisfaction problems. We show that, in general, the problem of finding a solution to an infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through a recursive tree. We also identify natural classes of infinite recursive constraint problems where the problem of finding a solution to the infinite recursive constraint problem is equivalent of the problem of finding an infinite path through a recursive finitely branching recursive trees or a recursive binary tree. There are a large number of results in the literature on the complexity of the problem of finding an infinite path through a recursive tree. Our main result allows us to automatically transfer such results to to give equivalent results about the complexity of the problem of finding a solution to a recursive CSP problem.

Tractable constraints on ordered domains

Artificial Intelligence, 1995

Finding solutions to a constraint satisfaction problem is known to be an NP-complete problem in general, but may be tractable in cases where either the set of allowed constraints or the graph structure is restricted. In this paper we identify a restricted set of contraints which gives rise to a class of tractable problems. This class generalizes the notion of a Horn formula in propositional logic to larger domain sizes. We give a polynomial time algorithm for solving such problems, and prove that the class of problems generated by any larger set of constraints is NP-complete.

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.

The complexity of quantified constraint satisfaction problems under structural restrictions

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.

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.

Complexity and polymorphisms for digraph constraint problems under some basic constructions

International Journal of Algebra and Computation, 2016

The role of polymorphisms in determining the complexity of constraint satisfaction problems is well established. In this context, we study the stability of CSP complexity and polymorphism properties under some basic graph theoretic constructions. As applications we observe a collapse in the applicability of algorithms for CSPs over directed graphs with both a total source and a total sink: the corresponding CSP is solvable by the “few subpowers algorithm” if and only if it is solvable by a local consistency check algorithm. Moreover, we find that the property of “strict width” and solvability by few subpowers are unstable under first-order reductions. The analysis also yields a complete characterization of the main polymorphism properties for digraphs whose symmetric closure is a complete graph.