New Sufficient Algebraic Conditions for Local Consistency over Homogeneous Structures of Finite Duality

TL;DR

This paper introduces new algebraic conditions that guarantee bounded width and tractability for certain infinite-domain CSPs, extending finite-domain algebraic characterizations to a broader class.

Contribution

It provides the first non-trivial algebraic conditions implying bounded width and tractability for a subclass of infinite-domain CSP templates within the Bodirsky-Pinsker conjecture.

Findings

Certain height 1 Maltsev conditions imply bounded width.

Bounded width leads to polynomial-time solvability.

Results apply to a broad class of templates, including previously unclassified ones.

Abstract

The path to the solution of Feder-Vardi dichotomy conjecture by Bulatov and Zhuk led through showing that more and more general algebraic conditions imply polynomial-time algorithms for the finite-domain Constraint Satisfaction Problems (CSPs) whose templates satisfy them. These investigations resulted in the discovery of the appropriate height 1 Maltsev conditions characterizing bounded strict width, bounded width, the applicability of the few-subpowers algorithm, and many others. For problems in the range of the similar Bodirsky-Pinsker conjecture on infinite-domain CSPs, one can only find such a characterization for the notion of bounded strict width, with a proof essentially the same as in the finite case. In this paper, we provide the first non-trivial results showing that certain height 1 Maltsev conditions imply bounded width, and in consequence tractability, for a natural subclass of templates within the Bodirsky-Pinsker conjecture which includes many templates in the literature as well as templates for which no complexity classification is known.