Strong subalgebras and the Constraint Satisfaction Problem

In 2007 it was conjectured that the Constraint Satisfaction Problem (CSP) over a constraint language $Γ$ is tractable if and only if $Γ$ is preserved by a weak near-unanimity (WNU) operation. After many efforts and partial results, this conjecture was independently proved by Andrei Bulatov and the author in 2017. In this paper we consider one of two main ingredients of my proof, that is, strong subalgebras that allow us to reduce domains of the variables iteratively. To explain how this idea works we show the algebraic properties of strong subalgebras and provide self-contained proof of two important facts about the complexity of the CSP. First, we prove that if a constraint language is not preserved by a WNU operation then the corresponding CSP is NP-hard. Second, we characterize all constraint languages that can be solved by local consistency checking. Additionally, we characterize all idempotent algebras not having a WNU term of a concrete arity $n$, not having a WNU term, having WNU terms of all arities greater than 2. Most of the results presented in the paper are not new, but I believe this paper can help to understand my approach to CSP and the new self-contained proof of known facts will be also useful.

💡 Research Summary

This paper revisits the celebrated CSP Dichotomy Conjecture, which states that a constraint satisfaction problem CSP(Γ) is tractable exactly when the constraint language Γ is preserved by a weak near‑unanimity (WNU) operation; otherwise it is NP‑complete. The author focuses on one of the two main ingredients of his independent proof of the conjecture (the other being Bulatov’s approach): the notion of strong subalgebras.

The work begins with a concise review of universal algebraic preliminaries—algebras, homomorphisms, the H, S, P operators, clones, relational clones, and the Galois correspondence between polymorphisms and invariant relations. This sets the stage for a purely algebraic treatment of CSP complexity, avoiding the heavy machinery of tame‑congruence theory that appears in earlier proofs.

In Section 3 the author defines a strong subuniverse (or strong subalgebra) of a finite idempotent algebra A as a subuniverse that belongs to one of three families:

1. Binary absorbing subuniverse – there exists a binary term operation t such that t(B,…,B,A,B,…,B)⊆B for any position of the argument A.
2. Central subuniverse – a binary absorbing subuniverse that additionally satisfies a non‑interference condition: for any a∉C, the pair (a,a) does not belong to the subalgebra generated by ( {a}×C )∪( C×{a} ).
3. PC (polynomially complete) subuniverse – a block of a congruence whose factor algebra is a direct product of algebras that are polynomially complete and lack non‑trivial binary absorbing, central, or projective subuniverses.

The author also introduces related concepts such as projective subuniverses, p‑affine algebras, and essential relations, and proves several lemmas that connect these notions to the existence (or non‑existence) of B‑essential relations.

The central technical result (Theorem 3.3) states that every finite idempotent algebra of size at least two must fall into one of five categories: (i) it has a non‑trivial binary absorbing subuniverse, (ii) it has a non‑trivial central subuniverse, (iii) it has a non‑trivial PC subuniverse, (iv) it admits a p‑affine quotient, or (v) it possesses a non‑trivial projective subuniverse. Moreover, if a projective subuniverse is present but not binary absorbing, the algebra must be essentially unary. This exhaustive classification is proved without recourse to deep term‑condition theorems; instead, the author relies on elementary clone arguments and the properties of essential relations.

Section 4 connects strong subalgebras to WNU operations. The author shows that the absence of a WNU term forces the existence of a WNU‑blocker, a configuration that prevents any strong subuniverse from existing. Conversely, when a WNU term of every arity > 2 is present, the algebraic structure guarantees a strong subuniverse for each variable domain. This dichotomy underlies the subsequent algorithmic treatment.

Section 5 applies the algebraic machinery to CSP complexity. The author first reduces any instance to its core (a minimal substructure preserving polymorphisms). Then, using the classification from Section 3, he proves two pivotal facts:

• Hardness – If Γ lacks a WNU polymorphism, the associated algebra has no strong subuniverse, and by constructing a suitable reduction (essentially the same as in