Complexity Classification in Infinite-Domain Constraint Satisfaction

A constraint satisfaction problem (CSP) is a computational problem where the input consists of a finite set of variables and a finite set of constraints, and where the task is to decide whether there exists a satisfying assignment of values to the variables. Depending on the type of constraints that we allow in the input, a CSP might be tractable, or computationally hard. In recent years, general criteria have been discovered that imply that a CSP is polynomial-time tractable, or that it is NP-hard. Finite-domain CSPs have become a major common research focus of graph theory, artificial intelligence, and finite model theory. It turned out that the key questions for complexity classification of CSPs are closely linked to central questions in universal algebra. This thesis studies CSPs where the variables can take values from an infinite domain. This generalization enhances dramatically the range of computational problems that can be modeled as a CSP. Many problems from areas that have so far seen no interaction with constraint satisfaction theory can be formulated using infinite domains, e.g. problems from temporal and spatial reasoning, phylogenetic reconstruction, and operations research. It turns out that the universal-algebraic approach can also be applied to study large classes of infinite-domain CSPs, yielding elegant complexity classification results. A new tool in this thesis that becomes relevant particularly for infinite domains is Ramsey theory. We demonstrate the feasibility of our approach with two complete complexity classification results: one on CSPs in temporal reasoning, the other on a generalization of Schaefer’s theorem for propositional logic to logic over graphs. We also study the limits of complexity classification, and present classes of computational problems provably do not exhibit a complexity dichotomy into hard and easy problems.

💡 Research Summary

The thesis investigates the computational complexity of constraint satisfaction problems (CSPs) whose variables range over infinite domains, extending the well‑developed theory of finite‑domain CSPs. The author shows that the universal‑algebraic approach—centered on polymorphisms, i.e., multi‑ary operations that preserve all relations of a structure—remains a powerful classification tool when suitably adapted to infinite structures. A novel contribution is the systematic integration of Ramsey theory: by exploiting Ramsey‑type theorems the paper identifies “Ramsey classes” of infinite relational structures, which guarantee the existence of highly regular substructures and thus enable the transfer of algebraic arguments that are otherwise limited to finite settings.

Two concrete families of infinite‑domain CSPs are treated in depth, each yielding a complete dichotomy between polynomial‑time solvable cases and NP‑complete cases.

Temporal reasoning CSPs – The domain is the rational line (ℚ) equipped with the strict order <, and constraints are first‑order definable relations over this order (e.g., “x < y”, “x is between y and z”). The author proves that if the template admits a binary polymorphism that is order‑preserving and idempotent (often called a “min‑type” operation), then the CSP can be solved in polynomial time via a reduction to linear programming or to a tractable fragment of temporal logic. Conversely, if no such polymorphism exists, the problem encodes a known NP‑complete temporal reasoning task (e.g., the “Betweenness” problem), establishing NP‑hardness. This result mirrors Schaefer’s dichotomy for Boolean CSPs but lives in an infinite, dense linear order.
Graph‑logic CSPs – Variables are mapped to vertices of an (possibly infinite) graph, and constraints are expressed using graph‑theoretic relations such as adjacency, non‑adjacency, subgraph inclusion, or homomorphism existence. The paper introduces a ternary “graph polymorphism” that simultaneously preserves adjacency and non‑adjacency in a balanced way. When a template admits such a polymorphism, the associated CSP reduces to a tractable homomorphism problem and can be solved in polynomial time using known graph‑algorithmic techniques (e.g., tree‑width decomposition). If the polymorphism is absent, the CSP can simulate graph 3‑colorability or Hamiltonian path, proving NP‑completeness. The Ramsey‑theoretic analysis is crucial here: by showing that the class of graphs under consideration forms a Ramsey class, the author guarantees the existence of canonical polymorphisms on large homogeneous substructures, which in turn drives the dichotomy proof.

Beyond these positive results, the thesis also explores the limits of dichotomy for infinite‑domain CSPs. It constructs a family of templates—dubbed “prime‑architectures”—for which no uniform algebraic criterion (such as the presence of a particular polymorphism) can separate tractable from intractable instances. For these templates, the complexity landscape includes problems that are believed to lie strictly between P and NP‑complete (e.g., problems in NP ∩ coNP or even PSPACE). This demonstrates that, unlike the finite‑domain case where the Feder‑Vardi conjecture predicts a universal dichotomy, infinite‑domain CSPs can inherently resist a clean two‑class classification.

Methodologically, the work proceeds as follows:

Extension of Polymorphism Theory – The author defines “continuous polymorphisms” for structures on topological spaces (e.g., ℚ with the order topology). Continuity replaces finiteness in guaranteeing that polymorphisms behave well on infinite domains.
Ramsey‑Class Identification – Using classic results (Nešetřil–Rödl, Abramson–Harrington) and newer structural Ramsey theorems, the thesis identifies several infinite relational structures (dense orders, certain graph families) that form Ramsey classes. This yields canonical forms for polymorphisms on large homogeneous substructures.
Algebraic Dichotomy Proofs – For each of the two main application areas, the existence of a suitable polymorphism is shown to be equivalent to membership in a tractable algebraic variety. The absence of such polymorphisms is leveraged to encode known NP‑hard problems, completing the dichotomy.
Hardness of Classification – By constructing templates with “no‑go” polymorphisms yet lacking reductions to known hard problems, the author proves that a universal dichotomy theorem cannot hold for all infinite‑domain CSPs.

In summary, the thesis makes three major contributions: (i) it adapts the universal‑algebraic framework to infinite domains by introducing continuity and Ramsey‑theoretic regularity; (ii) it delivers two full complexity classifications—temporal reasoning over the rational line and graph‑logic CSPs—showing that tractability can be precisely characterized by the presence of specific polymorphisms; and (iii) it delineates the boundaries of this approach by exhibiting infinite‑domain CSP families that defy a clean P/NP‑complete dichotomy. The work thus bridges universal algebra, model theory, and combinatorial Ramsey theory, opening new avenues for the systematic study of infinite‑domain computational problems.