Andrei A. Bulatov

A dichotomy theorem for constraint satisfaction problems on a 3-element set

The Constraint Satisfaction Problem (CSP) provides a common framework for many combinatorial problems. The general CSP is known to be NP-complete; however, certain restrictions on a possible form of constraints may affect the complexity and lead to tractable problem classes. There is, therefore, a fundamental research direction, aiming to separate those subclasses of the CSP that are tractable and those which remain NP-complete.Schaefer gave an exhaustive solution of this problem for the CSP on a 2-element domain. In this article, we generalise this result to a classification of the complexity of the CSP on a 3-element domain. The main result states that every subproblem of the CSP is either tractable or NP-complete, and the criterion separating them is that conjectured in Bulatov et al. [2005] and Bulatov and Jeavons [2001b]. We also characterize those subproblems for which standard constraint propagation techniques provide a decision procedure. Finally, we exhibit a polynomial time algorithm which, for a given set of allowed constraints, outputs if this set gives rise to a tractable problem class. To obtain the main result and the algorithm, we extensively use the algebraic technique for the CSP developed in Jeavons [1998b], Bulatov et al.[2005], and Bulatov and Jeavons [2001b].

Tractable conservative constraint satisfaction problems

In a constraint satisfaction problem (CSP), the aim is to find an assignment of values to a given set of variables, subject to specified constraints. The CSP is known to be NP-complete in general. However, certain restrictions on the form of the allowed constraints can lead to problems solvable in polynomial time. Such restrictions are usually imposed by specifying a constraint language. The principal research direction aims to distinguish those constraint languages, which give rise to tractable CSPs from those which do not. We achieve this goal for the widely used variant of the CSP, in which the set of values for each individual variable can be restricted arbitrarily. Restrictions of this type can be expressed by including in a constraint language all possible unary constraints. Constraint languages containing all unary constraints will be called conservative. We completely characterize conservative constraint languages that give rise to CSP classes solvable in polynomial time. In particular, this result allows us to obtain a complete description of those (directed) graphs H for which the List H-Coloring problem is polynomial time solvable.

A dichotomy theorem for constraints on a three-element set

The Constraint Satisfaction Problem (CSP) provides a common framework for many combinatorial problems. The general CSP is known to be NP-complete; however, certain restrictions on the possible form of constraints may affect the complexity, and lead to tractable problem classes. There is, therefore, a fundamental research direction, aiming to separate those subclasses of the CSP which are tractable, from those which remain NP-complete. In 1978 Schaefer gave an exhaustive solution of this problem for the CSP on a 2-element domain. In this paper we generalise this result to a classification of the complexity of CSPs on a 3-element domain. The main result states that every subclass of the CSP defined by a set of allowed constraints is either tractable or NP-complete, and the criterion separating them is that conjectured by Bulatov et al. (2001). We also exhibit a polynomial time algorithm which, for a given set of allowed constraints, determines whether if this set gives rise to a tractable problem class. To obtain the main result and the algorithm we extensively use the algebraic technique for the CSP developed by Jeavons (1998) and Bulatov et al.

The complexity of partition functions

We give a complexity theoretic classification of the counting versions of so-called H-colouring problems for graphs H that may have multiple edges between the same pair of vertices. More generally, we study the problem of computing a weighted sum of homomorphisms to a weighted graph H.The problem has two interesting alternative formulations: first, it is equivalent to computing the partition function of a spin system as studied in statistical physics. And second, it is equivalent to counting the solutions to a constraint satisfaction problem whose constraint language consists of two equivalence relations.In a nutshell, our result says that the problem is in polynomial time if the adjacency matrix of H has row rank 1, and #P-hard otherwise.

Constraint Satisfaction Problems and Finite Algebras

Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. In this paper we show that any restricted set of constraint types can be associated with a finite universal algebra. We explore how the computational complexity of a restricted constraint satisfaction problem is connected to properties of the corresponding algebra. Using these results we exhibit a common structural property of all known intractable constraint satisfaction problems. Finally, we classify all finite strictly simple surjective algebras with respect to tractability. The result is a dichotomy theorem which significantly generalises Schaefers dichotomy for the Generalised Satisfiability problem.

The complexity of the counting constraint satisfaction problem

The Counting Constraint Satisfaction Problem (#CSP(H)) over a finite relational structure H can be expressed as follows: given a relational structure G over the same vocabulary, determine the number of homomorphisms from G to H. In this article we characterize relational structures H for which (#CSP(H) can be solved in polynomial time and prove that for all other structures the problem is #P-complete.

Complexity of conservative constraint satisfaction problems

In a constraint satisfaction problem (CSP), the aim is to find an assignment of values to a given set of variables, subject to specified constraints. The CSP is known to be NP-complete in general. However, certain restrictions on the form of the allowed constraints can lead to problems solvable in polynomial time. Such restrictions are usually imposed by specifying a constraint language, that is, a set of relations that are allowed to be used as constraints. A principal research direction aims to distinguish those constraint languages that give rise to tractable CSPs from those that do not. We achieve this goal for the important version of the CSP, in which the set of values for each individual variable can be restricted arbitrarily. Restrictions of this type can be studied by considering those constraint languages which contain all possible unary constraints; we call such languages conservative. We completely characterize conservative constraint languages that give rise to polynomial time solvable CSP classes. In particular, this result allows us to obtain a complete description of those (directed) graphs H for which the List H-Coloring problem is solvable in polynomial time. The result, the solving algorithm, and the proofs heavily use the algebraic approach to CSP developed in Jeavons et al. [1997], Jeavons [1998], Bulatov et al. [2005], and Bulatov and Jeavons [2001b, 2003].

The complexity of maximal constraint languages

Many combinatorial search problems can be expressed as “constraint satisfaction problems” using an appropriate “constraint language”, that is, a set of relations over some fixed finite set of values. It is well-known that there is a trade-off between the expressive power of a constraint language and the complexity of the problems it can express. In the present paper we systematically study the complexity of all maximal constraint languages, that is, languages whose expressive power is just weaker than that of the language of all constraints. Using the algebraic invariance properties of constraints, we exhibit a strong necessary condition for tractability of such a constraint language. Moreover, we show that, at least for small sets of values, this condition is also sufficient.

The Complexity of the Counting Constraint Satisfaction Problem

The Counting Constraint Satisfaction Problem (

Quantified Constraints: Algorithms and Complexity

{\rm \#CSP}(\mathcal{H})