On tractability and congruence distributivity

Constraint languages that arise from finite algebras have recently been the object of study, especially in connection with the Dichotomy Conjecture of Feder and Vardi. An important class of algebras are those that generate congruence distributive varieties and included among this class are lattices, and more generally, those algebras that have near-unanimity term operations. An algebra will generate a congruence distributive variety if and only if it has a sequence of ternary term operations, called Jonsson terms, that satisfy certain equations. We prove that constraint languages consisting of relations that are invariant under a short sequence of Jonsson terms are tractable by showing that such languages have bounded relational width

Congruence modularity implies cyclic terms for finite algebras

An n-ary operation f : A(n) -> A is called cyclic if it is idempotent and f(a(1), a(2), a(3), ... , a(n)) = f(a(2), a(3), ... , a(n), a(1)) for every a(1), ... , a(n) is an element of A. We prove that every finite algebra A in a congruence modular variety has a p-ary cyclic term operation for any prime p greater than vertical bar A vertical bar

The Complexity of Quantified Constraint Satisfaction: Collapsibility, Sink Algebras, and the Three-Element Case

The constraint satisfaction probem (CSP) is a well-acknowledged framework in which many combinatorial search problems can be naturally formulated. The CSP may be viewed as the problem of deciding the truth of a logical sentence consisting of a conjunction of constraints, in front of which all variables are existentially quantified. The quantified constraint satisfaction problem (QCSP) is the generalization of the CSP where universal quantification is permitted in addition to existential quantification. The general intractability of these problems has motivated research studying the complexity of these problems under a restricted constraint language, which is a set of relations that can be used to express constraints. This paper introduces collapsibility, a technique for deriving positive complexity results on the QCSP. In particular, this technique allows one to show that, for a particular constraint language, the QCSP reduces to the CSP. We show that collapsibility applies to three known tractable cases of the QCSP that were originally studied using disparate proof techniques in different decades: Quantified 2-SAT (Aspvall, Plass, and Tarjan 1979), Quantified Horn-SAT (Karpinski, Kleine B\"{u}ning, and Schmitt 1987), and Quantified Affine-SAT (Creignou, Khanna, and Sudan 2001). This reconciles and reveals common structure among these cases, which are describable by constraint languages over a two-element domain. In addition to unifying these known tractable cases, we study constraint languages over domains of larger size

Relation identities in 3-distributive varieties

Let $\alpha$, $\beta$, $\gamma, \dots$ $\Theta$, $\Psi, \dots$ $R$, $S$, $T, \dots$ be variables for, respectively, congruences, tolerances and reflexive admissible relations. Let juxtaposition denote intersection. We show that if the identity $\alpha(\beta \circ \Theta) \subseteq \alpha \beta \circ \alpha \Theta \circ \alpha \beta$ holds in a variety $\mathcal {V}$, then $\mathcal {V}$ has a majority term, equivalently, $\mathcal {V}$ satisfies $\alpha (\beta \circ \gamma) \subseteq \alpha \beta \circ \alpha \gamma$. The result is unexpected, since in the displayed identity we have one more factor on the right and, moreover, if we let $\Theta$ be a congruence, we get a condition equivalent to $3$-distributivity, which is well-known to be strictly weaker than the existence of a majority term. The above result is optimal in many senses, for example, we show that slight variations on the displayed identity, such as $R (S \circ \gamma) \subseteq R S \circ R \gamma \circ R S$ or $R(S \circ T) \subseteq R S \circ RT \circ RT \circ RS$ hold in every $3$-distributive variety. Similar identities are valid even in varieties with $2$ Gumm terms, with no distributivity assumption. We also discuss relation identities in $n$-permutable varieties and present a few remarks about implication algebras.Comment: v2, entirely rewritten, the main theorems of v1 are now corollaries of more general results, v3, expanded the introduction, some further addition

Logic Column 17: A Rendezvous of Logic, Complexity, and Algebra

This article surveys recent advances in applying algebraic techniques to constraint satisfaction problems.Comment: 30 page

On tractability and congruence distributivity

Constraint languages that arise from finite algebras have recently been the object of study, especially in connection with the Dichotomy Conjecture of Feder and Vardi. An important class of algebras are those that generate congruence distributive varieties and included among this class are lattices, and more generally, those algebras that have near-unanimity term operations. An algebra will generate a congruence distributive variety if and only if it has a sequence of ternary term operations, called Jonsson terms, that satisfy certain equations. We prove that constraint languages consisting of relations that are invariant under a short sequence of Jonsson terms are tractable by showing that such languages have bounded relational width