# Constraint Satisfaction Problems over semilattice block Mal'tsev algebras

## TL;DR

This paper introduces a new hybrid algorithm for solving certain CSPs over semilattice block Mal'tsev algebras, demonstrating polynomial-time solvability for this class.

## Contribution

It presents a novel hybrid approach for CSPs over a restricted algebra class, expanding the solvable cases beyond previous methods.

## Key findings

- CSP over semilattice block Mal'tsev algebras is solvable in polynomial time.
- The new method effectively combines local propagation and basis generation techniques.
- The algebraic structure enables efficient solving of these CSPs.

## Abstract

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. We consider these method on a fairly restricted class of CSPs given by algebras we will call semilattice block Mal'tsev. An algebra A is called semilattice block Mal'tsev if it has a binary operation f, a ternary operation m, and a congruence s such that the quotient A/s with operation $f$ is a semilattice, $f$ is a projection on every block of s, and every block of s is a Mal'tsev algebra with Mal'tsev operation m. We show that the constraint satisfaction problem over a semilattice block Mal'tsev algebra is solvable in polynomial time.