# On the Scope of the Universal-Algebraic Approach to Constraint Satisfaction

## TL;DR

This paper extends the universal-algebraic approach to analyze the complexity of constraint satisfaction problems (CSPs) for infinite structures beyond omega-categorical cases, providing new criteria for hardness and polynomial-time solvability.

## Contribution

It generalizes key algebraic facts to broader infinite structures and applies these to characterize CSP complexity and solvability.

## Key findings

- Established general hardness criteria based on polymorphism absence.
- Provided a polymorphism-based characterization of first-order definable CSPs.
- Extended algebraic methods to non-omega-categorical infinite structures.

## Abstract

The universal-algebraic approach has proved a powerful tool in the study of the complexity of CSPs. This approach has previously been applied to the study of CSPs with finite or (infinite) omega-categorical templates, and relies on two facts. The first is that in finite or omega-categorical structures A, a relation is primitive positive definable if and only if it is preserved by the polymorphisms of A. The second is that every finite or omega-categorical structure is homomorphically equivalent to a core structure. In this paper, we present generalizations of these facts to infinite structures that are not necessarily omega-categorical. (This abstract has been severely curtailed by the space constraints of arXiv -- please read the full abstract in the article.) Finally, we present applications of our general results to the description and analysis of the complexity of CSPs. In particular, we give general hardness criteria based on the absence of polymorphisms that depend on more than one argument, and we present a polymorphism-based description of those CSPs that are first-order definable (and therefore can be solved in polynomial time).