Constraint Satisfaction Problems over Finitely Bounded Homogeneous Structures: a Dichotomy between FO and L-hard

TL;DR

This paper establishes a complexity dichotomy for CSPs over certain infinite structures, showing they are either first-order definable or L-hard, extending finite structure results to infinite cases.

Contribution

It proves a broad dichotomy for CSPs over finitely bounded homogeneous structures, generalizing finite structure results to infinite structures.

Findings

CSPs over these structures are either first-order definable or L-hard.

The paper generalizes Larose-Tesson theorem to infinite structures.

Provides a new proof technique for the dichotomy.

Abstract

Feder-Vardi conjecture, which proposed that every finite-domain Constraint Satisfaction Problem (CSP) is either in P or it is NP-complete, has been solved independently by Bulatov and Zhuk almost ten years ago. Bodirsky-Pinsker conjecture which states a similar dichotomy for countably infinite first-order reducts of finitely bounded homogeneous structures is wide open. In this paper, we prove that CSPs over first-order expansions of finitely bounded homogeneous model-complete cores are either first-order definable (and hence in non-uniform AC$^0$) or L-hard under first-order reduction. It is arguably the most general complexity dichotomy when it comes to the scope of structures within Bodirsky-Pinsker conjecture. Our strategy is that we first give a new proof of Larose-Tesson theorem, which provides a similar dichotomy over finite structures, and then generalize that new proof to infinite structures.