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

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.

💡 Research Summary

The paper addresses the complexity classification of constraint satisfaction problems (CSPs) defined over first‑order expansions of finitely bounded homogeneous model‑complete cores, a central class in the Bodirsky‑Pinsker conjecture. While the finite‑domain Feder‑Vardi conjecture has been resolved (Bulatov, Zhuk), the analogous dichotomy for countably infinite structures remains open. The authors prove a sharp dichotomy for the aforementioned class: every such CSP is either first‑order (FO) definable—hence belongs to non‑uniform AC⁰—or it is L‑hard under FO reductions. This result is the broadest known dichotomy within the scope of the Bodirsky‑Pinsker conjecture.

The proof proceeds in two main phases. First, the authors give a new proof of the Larose‑Tesson theorem for finite structures, which states that a finite‑domain CSP is either FO‑definable (and thus in AC⁰) or L‑hard. Their approach reframes CSP instances as relational structures to be homomorphically mapped into the template A, and leverages the concept of finite duality: a template has finite duality iff there exists a finite set of obstruction structures O such that a structure maps to A exactly when none of the obstructions map to it. Using Rossman’s preservation theorem, they connect finite duality with FO‑definability. Moreover, they show that if A is a core containing equality and all constants, then one can FO‑define a balanced implication, which yields a FO reduction from Graph Unreachability and thus L‑hardness. If such a balanced implication cannot be defined, the CSP is solvable by the (1‑)minimality (generalized arc‑consistency) algorithm.

The second phase lifts the argument to infinite, ω‑categorical structures. Let B be a finitely bounded homogeneous model‑complete core; such structures are ω‑categorical, k‑homogeneous, and ℓ‑bounded for some integers k, ℓ. The authors consider a first‑order expansion A of B (so A contains all relations of B). They introduce “A‑formulas”, which are first‑order formulas built only from the relational symbols of A (no equality unless present). Using tuples rather than single variables, they define (C, x̄, D, ȳ)‑implications: relational conditions linking two tuples of variables. If an A‑definable balanced implication exists, the same reduction from Graph Unreachability works, establishing L‑hardness.

If no balanced implication can be defined, the authors apply a (k, max(k,ℓ))‑minimality algorithm, a higher‑arity generalization of arc‑consistency. This algorithm produces “k‑tree A‑formulas”, tree‑shaped A‑formulas that capture all constraints of arity ≤ k that survive the minimality process. Either there are finitely many such k‑tree formulas witnessing unsatisfiable instances—yielding a finite set of obstructions and thus finite duality (hence FO‑definability)—or a large enough k‑tree formula can be transformed into a balanced implication, bringing us back to the L‑hard case. The proof carefully handles the infinite nature of B, ensuring that the construction of implications and obstruction sets respects the homogeneity and boundedness properties.

The paper provides illustrative examples. The dense linear order (ℚ,<) is a 2‑homogeneous, 3‑bounded model‑complete core; every first‑order expansion of it leads to L‑hard CSPs. Conversely, the countable universal homogeneous graph expanded with the non‑edge relation N yields a core whose base CSP is FO‑definable (in AC⁰), yet certain expansions that add relations defined by specific A‑formulas become L‑hard. These examples demonstrate how the presence or absence of A‑definable balanced implications determines the complexity class.

Significance: Theorem 1.3 (the main result) settles the dichotomy for the largest class of structures currently known within the Bodirsky‑Pinsker framework. By providing a proof technique that first re‑derives the finite‑domain dichotomy in a way amenable to infinite generalization, the authors open a pathway for future work: a new proof of the Bulatov‑Zhuk theorem that can be lifted to ω‑categorical settings could potentially resolve the full Bodirsky‑Pinsker conjecture. Moreover, the methodology clarifies the relationship between bounded width (solvability by minimality algorithms), finite duality, and L‑hardness, suggesting that similar techniques might classify CSPs into NL or L for broader families.

In summary, the paper establishes that for any first‑order expansion A of a finitely bounded homogeneous model‑complete core B, CSP(A) either admits a finite set of obstructions (hence FO‑definable and in AC⁰) or is L‑hard via a FO reduction. The proof hinges on the construction of A‑formulas, balanced implications, and k‑tree A‑formulas, extending finite‑domain techniques to infinite, ω‑categorical structures and providing a substantial advance toward the long‑standing Bodirsky‑Pinsker conjecture.