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

/pdf/the-complexity-of-the-counting-constraint-satisfaction-1wy2j4woyx.pdf

The Complexity of the Counting Constraint Satisfaction Problem

The Counting Constraint Satisfaction Problem (nCSP(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 (nCSP(H) can be solved in polynomial time and prove that for all other structures the problem is nP-complete.

/pdf/the-complexity-of-the-counting-constraint-satisfaction-gcikn9djet.pdf

The complexity of the counting constraint satisfaction problem

/pdf/lattice-varieties-covering-the-smallest-nonmodular-variety-37x3htuotv.pdf

Lattice varieties covering the smallest nonmodular variety.

The complexity of the counting constraint satisfaction problem.

/pdf/examples-of-locally-compact-noncompact-minimal-topological-4lavzsxpo3.pdf

Examples of locally compact noncompact minimal topological groups.

In studying a particular subdirectly irreducible lattice it is often important to know the location of a critical edge, that is, a quotient that is collapsed by every nontrivial congruence relation. We show that a critical edge can always be found in a rather natural location. In studying a particular subdirectly irreducible lattice L, it is often important to know the location of a critical edge of L, that is, a quotient u/v of L that is collapsed by every nontrivial congruence relation on L. Disregarding the trivial case in which L is distributive, and is therefore just a two-element chain, we shall show that such an edge can always be found in a rather natural location. A nondistributive lattice, of course, contains a sublattice isomorphic to the diamond A/3 or the pentagon N (see Figure 1). For distinct elements a, b and c of a lattice we write M (a, b, c) if a + b = b + c — c + a and ab = be = ca, and we write N(a, b, c) if be < a < c < a + b. Call a quotient u/v of L an M-quotient if it is an edge of a diamond, that is, if there exist a, b, c G L with M (a, b, c) such that either u/v = (a + tí)/ a or u/v = a/ab; call u/v an N-quotient if N(v, b, u) for some b G L.

/pdf/critical-edges-in-subdirectly-irreducible-lattices-1w2ssnz66v.pdf

Critical edges in subdirectly irreducible lattices

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Lattice varieties covering the smallest nonmodular variety.

Critical edges in subdirectly irreducible lattices