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

A CHR-based Implementation of Known Arc-Consistency

In classical CLP(FD) systems, domains of variables are completely known at the beginning of the constraint propagation process. However, in systems interacting with an external environment, acquiring the whole domains of variables before the beginning of constraint propagation may cause waste of computation time, or even obsolescence of the acquired data at the time of use. For such cases, the Interactive Constraint Satisfaction Problem (ICSP) model has been proposed as an extension of the CSP model, to make it possible to start constraint propagation even when domains are not fully known, performing acquisition of domain elements only when necessary, and without the need for restarting the propagation after every acquisition. In this paper, we show how a solver for the two sorted CLP language, defined in previous work, to express ICSPs, has been implemented in the Constraint Handling Rules (CHR) language, a declarative language particularly suitable for high level implementation of constraint solvers.Comment: 22 pages, 2 figures, 1 table To appear in Theory and Practice of Logic Programming (TPLP

Searching for an algebra on CSP solutions

The Constraint Satisfaction Problem (CSP) is a problem of computing a homomorphism ${\mathbf R}\to {\bf \Gamma}$ between two relational structures, where ${\mathbf R}$ is defined over a domain $V$ and ${\bf \Gamma}$ is defined over a domain $D$. In a fixed template CSP, denoted $CSP({\bf \Gamma})$, the right side structure ${\bf \Gamma}$ is fixed and the left side structure ${\mathbf R}$ is unconstrained. We consider the following problem: given a prespecified finite set of algebras ${\mathcal B}$ whose domain is $D$, is it possible to present the solutions set of a given instance of $CSP({\bf \Gamma})$ (which is an input to the problem) as a subalgebra of ${\mathbb A}_1\times ... \times {\mathbb A}_{|V|}$ where ${\mathbb A}_i\in {\mathcal B}$? We study this problem and show that it can be reformulated as an instance of a certain fixed-template CSP, over another template ${\bf \Gamma}^{\mathcal B}$. First, we demonstrate examples of ${\mathcal B}$ for which $CSP({\bf \Gamma}^{\mathcal B})$ is tractable for any, possibly NP-hard, ${\bf \Gamma}$. Under natural assumptions on ${\mathcal B}$, we prove that $CSP({\bf \Gamma}^{\mathcal B})$ can be reduced to a certain fragment of $CSP({\bf \Gamma})$. We also study the conditions under which $CSP({\bf \Gamma})$ can be reduced to $CSP({\bf \Gamma}^{\mathcal B})$. Since the complexity of $CSP({\bf \Gamma}^{\mathcal B})$ is defined by $Pol({\bf \Gamma}^{\mathcal B})$, we study the relationship between $Pol({\bf \Gamma})$ and $Pol({\bf \Gamma}^{\mathcal B})$. It turns out that if $\mathcal{B}$ is preserved by $p\in Pol({\bf \Gamma})$, then $p$ can be extended to a polymorphism of ${\bf \Gamma}^{\mathcal B}$. In the end to demonstrate usefulness of our definitions we study one case when ${\bf \Gamma}$ is not of bounded width, but ${\bf \Gamma}^{\mathcal B}$ is of bounded width (i.e. has a richer structure of polymorphisms).Comment: 34 page

Aggregation of Votes with Multiple Positions on Each Issue

We consider the problem of aggregating votes cast by a society on a fixed set of issues, where each member of the society may vote for one of several positions on each issue, but the combination of votes on the various issues is restricted to a set of feasible voting patterns. We require the aggregation to be supportive, i.e. for every issue $j$ the corresponding component $f_j$ of every aggregator on every issue should satisfy $f_j(x_1, ,\ldots, x_n) \in \{x_1, ,\ldots, x_n\}$. We prove that, in such a set-up, non-dictatorial aggregation of votes in a society of some size is possible if and only if either non-dictatorial aggregation is possible in a society of only two members or a ternary aggregator exists that either on every issue $j$ is a majority operation, i.e. the corresponding component satisfies $f_j(x,x,y) = f_j(x,y,x) = f_j(y,x,x) =x, \forall x,y$, or on every issue is a minority operation, i.e. the corresponding component satisfies $f_j(x,x,y) = f_j(x,y,x) = f_j(y,x,x) =y, \forall x,y.$ We then introduce a notion of uniformly non-dictatorial aggregator, which is defined to be an aggregator that on every issue, and when restricted to an arbitrary two-element subset of the votes for that issue, differs from all projection functions. We first give a characterization of sets of feasible voting patterns that admit a uniformly non-dictatorial aggregator. Then making use of Bulatov's dichotomy theorem for conservative constraint satisfaction problems, we connect social choice theory with combinatorial complexity by proving that if a set of feasible voting patterns $X$ has a uniformly non-dictatorial aggregator of some arity then the multi-sorted conservative constraint satisfaction problem on $X$, in the sense introduced by Bulatov and Jeavons, with each issue representing a sort, is tractable; otherwise it is NP-complete