Constraint Satisfaction Problems over semilattice block Mal'tsev algebras

Abstract

The Dichotomy Conjecture for the Constraint Satisfaction Problem (CSP) was recently settled, independently by Zhuk and the author. The proofs of this conjecture are rather sophisticated and require deep understanding of the algebraic structure of CSPs. This paper is a precursor of the author's proof of the Dichotomy Conjecture, and represents its main ideas in a simpler and clearer form in a more restricted class of the CSP.There are two well-known types of algorithms for solving CSPs: local propagation and generating a basis of the solution space. For several years the focus of the CSP research has been on ‘hybrid’ algorithms that somehow combine the two approaches. In this paper we present a new method of such hybridization that allows us to solve certain CSPs that has been out of reach for a quite a while, and eventually leads to resolving the Dichotomy Conjecture.We apply this method to CSPs parametrized by a universal algebra, an approach that has been very popular in the last decade or so. Specifically, we consider a fairly restricted class of algebras we will call semilattice block Mal'tsev. An algebra A is semilattice block Mal'tsev if it has a binary operation f, a ternary operation m, and a congruence σ such that the quotient Aσ with operation f is a semilattice, f is a projection on every block of σ, and every block of σ is a Mal'tsev algebra with Mal'tsev operation m. This means that the domain in such a CSP is partitioned into blocks such that if the problem is considered on the quotient set Aσ, it can be solved by a simple constraint propagation algorithm. On the other hand, if the problem is restricted on individual σ-blocks, it can be solved by generating a basis of the solution space. We show that the two methods can be combined in a highly nontrivial way, and therefore the constraint satisfaction problem over a semilattice block Mal'tsev algebra is solvable in polynomial time.