Topology is irrelevant (in a dichotomy conjecture for infinite domain constraint satisfaction problems)

TL;DR

This paper proves that the complexity boundary for certain infinite domain constraint satisfaction problems can be characterized solely by an algebraic identity, removing the need for topological considerations.

Contribution

It demonstrates that the tractability boundary for infinite domain CSPs is characterized by a specific algebraic identity, simplifying the understanding of the problem's complexity.

Findings

The complexity boundary is characterized by the pseudo-Siggers identity.

Topological properties are not essential in the algebraic characterization.

Provides a purely algebraic criterion for a topological property of $oldsymbol{ extomega}$-categorical structures.

Abstract

The tractability conjecture for finite domain Constraint Satisfaction Problems (CSPs) stated that such CSPs are solvable in polynomial time whenever there is no natural reduction, in some precise technical sense, from the 3-SAT problem; otherwise, they are NP-complete. Its recent resolution draws on an algebraic characterization of the conjectured borderline: the CSP of a finite structure permits no natural reduction from 3-SAT if and only if the stabilizer of the polymorphism clone of the core of the structure satisfies some nontrivial system of identities, and such satisfaction is always witnessed by several specific nontrivial systems of identities which do not depend on the structure. The tractability conjecture has been generalized in the above formulation to a certain class of infinite domain CSPs, namely, CSPs of reducts of finitely bounded homogeneous structures. It was subsequently shown that the conjectured borderline between hardness and tractability, i.e., a natural reduction from 3-SAT, can be characterized for this class by a combination of algebraic and topological properties. However, it was not known whether the topological component is essential in this characterization. We provide a negative answer to this question by proving that the borderline is characterized by one specific algebraic identity, namely the pseudo-Siggers identity $\alpha s(x,y,x,z,y,z) \approx \beta s(y,x,z,x,z,y)$. This accomplishes one of the steps of a proposed strategy for reducing the infinite domain CSP dichotomy conjecture to the finite case. Our main theorem is also of independent mathematical interest, characterizing a topological property of any $\omega$-categorical core structure (the existence of a continuous homomorphism of a stabilizer of its polymorphism clone to the projections) in purely algebraic terms (the failure of an identity as above).