The complexity and approximability of the constraint satisfaction problem (CSP) has been actively studied over the past 20 years. A new version of the CSP, the promise CSP (PCSP), has recently been proposed, motivated by open questions about the approximability of variants of satisfiability and graph colouring. The PCSP significantly extends the standard decision CSP. The complexity of CSPs with a fixed constraint language on a finite domain has recently been fully classified, greatly guided by the algebraic approach, which uses polymorphisms—high-dimensional symmetries of solution spaces—to analyse the complexity of problems. The corresponding classification for PCSPs is wide open and includes some long-standing open questions, such as the complexity of approximate graph colouring, as special cases. The basic algebraic approach to PCSP was initiated by Brakensiek and Guruswami, and in this article, we significantly extend it and lift it from concrete properties of polymorphisms to their abstract properties. We introduce a new class of problems that can be viewed as algebraic versions of the (Gap) Label Cover problem and show that every PCSP with a fixed constraint language is equivalent to a problem of this form. This allows us to identify a “measure of symmetry” that is well suited for comparing and relating the complexity of different PCSPs via the algebraic approach. We demonstrate how our theory can be applied by giving both general and specific hardness/tractability results. Among other things, we improve the state-of-the-art in approximate graph colouring by showing that, for any k ≥ 3, it is NP-hard to find a (2 k -1)-colouring of a given k -colourable graph.

A fundamental fact for the algebraic theory of constraint satisfaction problems (CSPs) over a fixed template is that pp-interpretations between at most countable ω-categorical relational structures have two algebraic counterparts for their polymorphism clones: a semantic one via the standard algebraic operators H, S, P, and a syntactic one via clone homomorphisms (capturing identities). We provide a similar characterization which incorporates all relational constructions relevant for CSPs, that is, homomorphic equivalence and adding singletons to cores in addition to pp-interpretations. For the semantic part we introduce a new construction, called reflection, and for the syntactic part we find an appropriate weakening of clone homomorphisms, called h1 clone homomorphisms (capturing identities of height 1).As a consequence, the complexity of the CSP of an at most countable ω-categorical structure depends only on the identities of height 1 satisfied in its polymorphism clone as well as the natural uniformity thereon. This allows us in turn to formulate a new elegant dichotomy conjecture for the CSPs of reducts of finitely bounded homogeneous structures.Finally, we reveal a close connection between h1 clone homomorphisms and the notion of compatibility with projections used in the study of the lattice of interpretability types of varieties.

The complexity and approximability of the constraint satisfaction problem (CSP) has been actively studied over the last 20 years. A new version of the CSP, the promise CSP (PCSP) has recently been proposed, motivated by open questions about the approximability of variants of satisfiability and graph colouring. The PCSP significantly extends the standard decision CSP. The complexity of CSPs with a fixed constraint language on a finite domain has recently been fully classified, greatly guided by the algebraic approach, which uses polymorphisms -high-dimensional symmetries of solution spaces -to analyse the complexity of problems. The corresponding classification for PCSPs is wide open and includes some long-standing open questions, such as the complexity of approximate graph colouring, as special cases.The basic algebraic approach to PCSP was initiated by Brakensiek and Guruswami, and in this paper we significantly extend it and lift it from concrete properties of polymorphisms to their abstract properties. We introduce a new class of problems that can be viewed as algebraic versions of the (Gap) Label Cover problem, and show that every PCSP with a fixed constraint language is equivalent to a problem of this form. This allows us to identify a "measure of symmetry" that is well suited for comparing and relating the complexity of different PCSPs via the algebraic approach. We demonstrate how our theory can be applied by improving the state-of-the-art in approximate graph colouring: we show that, for any k ≥ 3, it is NP-hard to find a (2k − 1)-colouring of a given k-colourable graph. CCS CONCEPTS• Theory of computation → Problems, reductions and completeness.

A. We study the complexity of approximation on satis able instances for graph homomorphism problems. For a xed graph H, the H-colouring problem is to decide whether a given graph has a homomorphism to H. By a result of Hell and Nešetřil, this problem is NP-hard for any non-bipartite graph H. In the context of promise constraint satisfaction problems, Brakensiek and Guruswami conjectured that this hardness result extends to promise graph homomorphism as follows: x any non-bipartite graph H and another graph G with a homomorphism from H to G, it is NP-hard to nd a homomorphism to G from a given H-colourable graph. Arguably, the two most important special cases of this conjecture are when H is xed to be the complete graph on 3 vertices (and G is any graph with a triangle) and when G is the complete graph on 3 vertices (and H is any 3-colourable graph). The former case is equivalent to the notoriously di cult approximate graph colouring problem. In this paper, we con rm the Brakensiek-Guruswami conjecture for the latter case. Our proofs rely on a novel combination of the universal-algebraic approach to promise constraint satisfaction, that was recently developed by Barto, Bulín and the authors, with some ideas from algebraic topology.

A. The approximate graph colouring problem concerns colouring a k-colourable graph with c colours, where c ≥ k. This problem naturally generalises to promise graph homomorphism and further to promise constraint satisfaction problems. Complexity analysis of all these problems is notoriously di cult. In this paper, we introduce two new techniques to analyse the complexity of promise CSPs: one is based on topology and the other on adjunction. We apply these techniques, together with the previously introduced algebraic approach, to obtain new NP-hardness results for a signi cant class of approximate graph colouring and promise graph homomorphism problems.