Separation of congruence intervals and implications
Andrei A. Bulatov
TL;DR
The paper develops an algebraic framework for finite idempotent algebras omitting type 1 to advance the CSP dichotomy program. It introduces separation of congruence intervals and collapsing polynomials, and couples these with centralizers, alignment, quasi-decomposition, and chaining to analyze subdirect products via edge-colored graphs. The main contributions include symmetric- and relative-symmetry results for separation, the Collapsing/Congruence Lemma, and a robust chaining-maximality toolkit that constrains how factor blocks interact. Together, these tools underpin a structural program aimed at decomposing CSP instances into tractable, modular components, contributing to the algebraic proof strategy for the Feder–Vardi Dichotomy Theorem.
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).