Notes on CSPs and Polymorphisms
Zarathustra Brady
TL;DR
This work surveys the algebraic CSP framework, framing CSPs as homomorphism problems and connecting constraint templates to clone theory via the Inv–Pol Galois duality. It highlights core concepts such as few subpowers, absorbing subalgebras, and bounded width, and presents key algorithmic dichotomies for conservative and Taylor-type algebras, including the use of pp-definability, HSP/Birkhoff theory, and the core/reduct perspective. The paper also discusses extensions to valued and promise CSPs, the role of multisorted CSPs, and the modern dichotomy results that classify finite templates into polynomial-time solvable and NP-hard cases, with implications for tractable algorithm design. Overall, it provides a comprehensive algebraic lens on CSP complexity, uniting structure theory and computation with broad implications for constraint reasoning and optimization, all while emphasizing the central role of polymorphisms and their identities in determining tractability.
Abstract
These are notes from a multi-year learning seminar on the algebraic approach to Constraint Satisfaction Problems (CSPs). The main topics covered are the theory of algebraic structures with few subpowers, the theory of absorbing subalgebras and its applications to studying CSP templates which can be solved by local consistency methods, and the dichotomy theorem for conservative CSP templates. Subsections and appendices cover supplementary material.