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.

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.

It is well known that the constraint satisfaction problem over general relational structures can be reduced in polynomial time to digraphs. We present a simple variant of such a reduction and use it to show that the algebraic dichotomy conjecture is equivalent to its restriction to digraphs and that the polynomial reduction can be made in logspace. We also show that our reduction preserves the bounded width property, i.e., solvability by local consistency methods. We discuss further algorithmic properties that are preserved and related open problems. The main resultsIn general, fixed template CSPs can be modelled as relational structure homomorphism problems [9]. For detailed formal definitions of relational structures, homomorphisms and polymorphisms, see Section 3.Let A be a finite structure with signature R (the fixed template), then the constraint satisfaction problem for A is the following decision problem. Constraint satisfaction problem for A. CSP(A)INSTANCE: A finite R-structure X. QUESTION: Is there a homomorphism from X to A?The dichotomy conjecture [9] can be stated as follows:CSP dichotomy conjecture. Let A be a finite relational structure. Then CSP(A) is solvable in polynomial time or NP-complete.The dichotomy conjecture is equivalent to its restriction to digraphs [9], and thus can be restated as follows:CSP dichotomy conjecture. Let H be a finite digraph. Then CSP(H) is solvable in polynomial time or NP-complete.Every finite relational structure A has a unique core substructure A ′ (see Section 3.3 for the precise definition) such that CSP(A) and CSP(A ′ ) are identical problems, i.e., the "yes" and "no" instances are precisely the same. The algebraic dichotomy conjecture [6] is the following:Algebraic CSP dichotomy conjecture. Let A be a finite relational structure that is a core. If A has a Taylor polymorphism then CSP(A) is solvable in polynomial time, otherwise CSP(A) is NP-complete.Indeed, perhaps the above conjecture should be called the algebraic tractability conjecture since it is known that if a core A does not possess a Taylor polymorphism, then CSP(A) is NP-complete [6].Feder and Vardi [9] proved that every fixed template CSP is polynomial time equivalent to a digraph CSP. This article will provide the following theorem, which replaces "polynomial time" with "logspace" and reduces the algebraic dichotomy conjecture to digraphs.

Abstract. It is well known that the constraint satisfaction problem over a general relational structure A is polynomial time equivalent to the constraint problem over some associated digraph. We present a variant of this construction and show that the corresponding constraint satisfaction problem is logspace equivalent to that over A. Moreover, we show that almost all of the commonly encountered polymorphism properties are held equivalently on the A and the constructed digraph. As a consequence, the Algebraic CSP dichotomy conjecture as well as the conjectures characterizing CSPs solvable in logspace and in nondeterministic logspace are equivalent to their restriction to digraphs.

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