Conservative constraint satisfaction re-revisited

Conservative constraint satisfaction problems (CSPs) constitute an important particular case of the general CSP, in which the allowed values of each variable can be restricted in an arbitrary way. Problems of this type are well studied for graph homomorphisms. A dichotomy theorem characterizing conservative CSPs solvable in polynomial time and proving that the remaining ones are NP-complete was proved by Bulatov in 2003. Its proof, however, is quite long and technical. A shorter proof of this result based on the absorbing subuniverses technique was suggested by Barto in 2011. In this paper we give a short elementary prove of the dichotomy theorem for the conservative CSP

Galois correspondence for counting quantifiers

We introduce a new type of closure operator on the set of relations, max-implementation, and its weaker analog max-quantification. Then we show that approximation preserving reductions between counting constraint satisfaction problems (#CSPs) are preserved by these two types of closure operators. Together with some previous results this means that the approximation complexity of counting CSPs is determined by partial clones of relations that additionally closed under these new types of closure operators. Galois correspondence of various kind have proved to be quite helpful in the study of the complexity of the CSP. While we were unable to identify a Galois correspondence for partial clones closed under max-implementation and max-quantification, we obtain such results for slightly different type of closure operators, k-existential quantification. This type of quantifiers are known as counting quantifiers in model theory, and often used to enhance first order logic languages. We characterize partial clones of relations closed under k-existential quantification as sets of relations invariant under a set of partial functions that satisfy the condition of k-subset surjectivity. Finally, we give a description of Boolean max-co-clones, that is, sets of relations on {0,1} closed under max-implementations.Comment: 28 pages, 2 figure

Separation of congruence intervals and implications

The Constraint Satisfaction Problem (CSP) has been intensively studied in many areas of computer science and mathematics. The approach to the CSP based on tools from universal algebra turned out to be the most successful one to study the complexity and algorithms for this problem. Several techniques have been developed over two decades. One of them is through associating edge-colored graphs with algebras and studying how the properties of algebras are related with the structure of the associated graphs. This approach has been introduced in our previous two papers (A.Bulatov, Local structure of idempotent algebras I,II. arXiv:2006.09599, arXiv:2006.10239, 2020). In this paper we further advance it by introducing new structural properties of finite idempotent algebras omitting type 1 such as separation congruences, collapsing polynomials, and their implications for the structure of subdirect products of finite algebras. This paper also provides the algebraic background for our proof of Feder-Vardi Dichotomy Conjecture (A. Bulatov, A Dichotomy Theorem for Nonuniform CSPs. FOCS 2017: 319-330).Comment: arXiv admin note: substantial text overlap with arXiv:1703.0302