Equations in oligomorphic clones and the Constraint Satisfaction Problem for $\omega$-categorical structures

TL;DR

This paper investigates the algebraic conditions that determine the complexity of constraint satisfaction problems (CSPs) for certain infinite structures, establishing equivalences and applying Ramsey theory to advance understanding.

Contribution

It proves the equivalence of two major conjectures on CSP tractability for $\omega$-categorical structures and introduces new methods using linear identities and Ramsey properties.

Findings

Equivalence of two CSP conjectures for $\omega$-categorical structures.

Linear identities imply orbit growth constraints in automorphism groups.

New proof that $\omega$-categorical structures are homomorphically equivalent to model-complete cores.

Abstract

There exist two conjectures for constraint satisfaction problems (CSPs) of reducts of finitely bounded homogeneous structures: the first one states that tractability of the CSP of such a structure is, when the structure is a model-complete core, equivalent to its polymorphism clone satisfying a certain non-trivial linear identity modulo outer embeddings. The second conjecture, challenging the approach via model-complete cores by reflections, states that tractability is equivalent to the linear identities (without outer embeddings) satisfied by its polymorphisms clone, together with the natural uniformity on it, being non-trivial. We prove that the identities satisfied in the polymorphism clone of a structure allow for conclusions about the orbit growth of its automorphism group, and apply this to show that the two conjectures are equivalent. We contrast this with a counterexample showing that $\omega$-categoricity alone is insufficient to imply the equivalence of the two conditions above in a model-complete core. Taking a different approach, we then show how the Ramsey property of a homogeneous structure can be utilized for obtaining a similar equivalence under different conditions. We then prove that any polymorphism of sufficiently large arity which is totally symmetric modulo outer embeddings of a finitely bounded structure can be turned into a non-trivial system of linear identities, and obtain non-trivial linear identities for all tractable cases of reducts of the rational order, the random graph, and the random poset. Finally, we provide a new and short proof, in the language of monoids, of the theorem stating that every $\omega$-categorical structure is homomorphically equivalent to a model-complete core.