Separation of congruence intervals and implications

TL;DR

This paper advances the algebraic approach to the CSP by introducing new structural properties of finite idempotent algebras, with implications for understanding subdirect products and contributing to the proof of the Feder-Vardi Dichotomy Conjecture.

Contribution

It introduces separation congruences and collapsing polynomials, deepening the algebraic analysis of CSPs and their complexity classifications.

Findings

Introduced separation congruences for idempotent algebras

Analyzed implications for subdirect product structures

Supported proof of the Feder-Vardi Dichotomy Conjecture

Abstract

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).