Many combinatorial search problems can be expressed as `constraint satisfaction problems', and this class of problems is known to be NP-complete in general. In this paper we investigate the subclasses which arise from restricting the possible constraint types. We first show that any set of constraints which does not give rise to an NP-complete class of problems must satisfy a certain type of algebraic closure condition. We then investigate all the different possible forms of this algebraic closure property, and establish which of these are sufficient to ensure tractability. As examples, we show that all known classes of tractable constraints over finite domains can be characterised by such an algebraic closure property. Finally, we describe a simple computational procedure which can be used to determine the closure properties of a given set of constraints. This procedure involves solving a particular constraint satisfaction problem, which we call an `indicator problem'. Keywords: Cons...

Conjunctive-query containment is recognized as a fundamental problem in database query evaluation and optimization. At the same time, constraint satisfaction is recognized as a fundamental problem in artificial intelligence. What do conjunctive-query containment and constraint satisfaction have in common? Our main conceptual contribution in this paper is to point out that, despite their very different formulation, conjunctive-query containment and constraint satisfaction are essentially the same problem. The reason is that they can be recast as the following fundamental algebraic problem: given two finite relational structures A and B, is there a homomorphism h : A ! B? As formulated above, the homomorphism problem is uniform in the sense that both relational structures A and B are part of the input. By fixing the structure B, one obtains the following non-uniform problem: given a finite relational structure A, is there a homomorphism h : A ! B? In general, non-uniform tractability results do not uniformize. Thus, it is natural to ask: which tractable cases of non-uniform tractability results for constraint satisfaction and conjunctive-query containment do uniformize? Our main technical contribution in this paper is to show that several cases of tractable non-uniform constraint satisfaction problems do indeed uniformize. We exhibit three non-uniform tractability results that uniformize and, thus, give rise to polynomial-time solvable cases of constraint satisfaction and conjunctive-query containment.

Abstract. Let H be a fixed digraph. We consider the H-colouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the absence in G of certain tree-like obstructions. Specifically, we say that H has tree duality if, for all digraphs G, G is not homomorphic to H if and only if there is an oriented tree which is homomorphic to G but not to H. Weprovethatif Hhas tree duality then the H-colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the X-property studied by Gutjahr, Welzl, and Woeginger. We then focus on the case when H itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads H for which the H-colouring problem is NP-complete. We contrast these with several families of oriented triads H which have tree duality, or bounded treewidth duality, and hence polynomial H-colouring problems. If P � = NP, then no oriented triad H with an NP-complete H-colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad H. We prove that none of the oriented triads H with NP-complete Hcolouring problems given in the companion paper has tree duality. 1.

. We introduce in this paper the notion of the chromatic number of an oriented graph G (that is of an antisymmetric directed graph) dened as the minimum order of an oriented graph H such that G admits a homomorphism to H . We study the chromatic number of oriented k-trees and of oriented graphs with bounded degree. We show that there exist oriented k-trees with chromatic number at least 2 k+1 1 and that every oriented k-tree has chromatic number at most (k + 1) 2 k . For 2-trees and 3-trees we decrease these upper bounds respectively to 7 and 16 and show that these new bounds are tight. As a particular case, we obtain that oriented outerplanar graphs have chromatic number at most 7 and that this bound is tight too. We then show that every oriented graph with maximum degree k has chromatic number at most (2k 1) 2 2k 2 . For oriented graphs with maximum degree 2 we decrease this bound to 5 and show that this new bound is tight. For oriented graphs with maximum degree 3 we decrease this bound to 16 and conjecture that there exists no such connected graph with chromatic number greater than 7. Keywords. Graph coloring, Graph homomorphism, Oriented coloring. 1

Although the constraint satisfaction problem is NP-complete in general, a number of constraint classes have been identified for which some fixed level of local consistency is sufficient to ensure global consistency. In this paper, we describe a simple algebraic property which characterises all possible constraint types for which strong k-consistency is sufficient to ensure global consistency, for each k ? 2. We give a number of examples to illustrate the application of this result. 1 Introduction The constraint satisfaction problem provides a framework in which it is possible to express, in a natural way, many combinatorial problems encountered in artificial intelligence and elsewhere. The aim in a constraint satisfaction problem is to find an assignment of values to a given set of variables subject to constraints on the values which can be assigned simultaneously to certain specified subsets of variables. The constraint satisfaction problem is known to be an NP-complete problem in ge...

In this paper we study the approximability of boolean constraint satisfaction problems. A problem in this class consists of some collection of "constraints " (i.e., functions f: f0; 1g k! f0; 1g); an instance of a problem

In this report we discuss constraint satisfaction problems. These are problems in which values must be assigned to a collection of variables, subject to specified constraints. We focus specifically on problems in which the domain of possible values for each variable is finite. The report surveys the various conditions that have been shown to be sufficient to ensure tractability in these problems. These are broken down into three categories: ffl Conditions on the overall structure; ffl Conditions on the nature of the constraints; ffl Conditions on bounded pieces of the problem. 1 Introduction A constraint satisfaction problem is a way of expressing simultaneous requirements for values of variables. The study of constraint satisfaction problems was initiated by Montanari in 1974 [34], when he used them as a way of describing certain combinatorial problems arising in image-processing. It was quickly realised that the same general framework was applicable to a much wider class of probl...

We show that for every conjunctive query, the complexity of evaluating it on a probabilistic database is either PTIME or #P-complete, and we give an algorithm for deciding whether a given conjunctive query is PTIME or #P-complete. The dichotomy property is a fundamental result on query evaluation on probabilistic databases and it gives a complete classification of the complexity of conjunctive queries. 1. PROBLEM STATEMENT Fix a relational vocabulary R1,..., Rk, denoted R. A tuple-independent probabilistic structure is a pair (A, p) where A = (A, R A 1,..., R A k) is first order structure and p is a function that associates to each tuple t in A a rational number p(t) ∈ [0, 1]. A probabilistic structure (A,p) induces a probability distribution on the set of substructures B of A by: p(B) = kY ( Y p(t) × i=1 t∈RB i Y t∈R A i −RB i (1 − p(t))) (1) where B ⊆ A, more precisely B = (A, R B 1,..., B B k) is s.t. R B i ⊆ R A i for i = 1, k. A conjunctive query, q, is a sentence of the form ∃¯x.(ϕ1 ∧... ∧ϕm), where each ϕi is a positive atomic predicate R(t), called a sub-goal, and the tuple t consists of variables and/or constants. As usual, we drop the existential quantifiers and the ∧, writing q = ϕ1, ϕ2,..., ϕm. A conjunctive property is a property on structures defined by a conjunctive query q, and its probability on a probabilistic structure (A, p) is defined as: p(q) = X p(B) (2)

List homomorphisms generalize list colourings in the following way: Given graphs G; H , and lists L(v) ` V (H); v 2 V (G), a list homomorphism of G to H with respect to the lists L is a mapping f : V (G) ! V (H) such that uv 2 E(G) implies f(u)f(v) 2 E(H), and f(v) 2 L(v) for all v 2 V (G). The list homomorphism problem for a fixed graph H asks whether or not an input graph G together with lists L(v) ` V (H), v 2 V (G), admits a list homomorphism with respect to L. The list homomorphism problem was introduced by Feder and Hell, who proved that for reflexive graphs H (that is, for graphs H in which every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP-complete otherwise. Here we consider graphs H without loops, and find that the problem is closely related to circular arc graphs. We show that the list homomorphism problem is polynomial time solvable if the complement of H is a circular arc graph of clique covering number two, and is NP-complete otherwise. For the purposes of the proof we give a new characterization of circular arc graphs of clique covering number two, by the absence of a structure analogous to Gallai's asteroids. Both results point to a surprising similarity between interval graphs and the complements of circular arc graphs of clique covering number two. Key Words: Homomorphisms, list-homomorphisms, retractions, asteroidal triples, circular arc graphs, algorithms, complexity. 1

Schaefer showed, long ago, that there are, essentially, only three non-trivial classes of conjunctive Boolean formulae (or constraint satisfaction problems) for which satis ability can be decided in polynomial time (assuming P 6= NP ). These three classes are LIN, 2-SAT and HORN-SAT. LIN is the constraint satisfaction problem in which all the constraints are linear equations modulo 2. 2-SAT is the constraint satisfaction problem in which all the constraints are disjunctions of at most two variables or their negations. HORN-SAT is the constraint satisfaction problem in which all the constraints are Horn clauses, i.e., disjunctions containing at most one negated variable.