Abstract. The quantified constraint satisfaction problem is the problem of evaluating a sentence with both quantifiers, over relations from some constraint language, with the conjunction as the only connective. We show that for any constraint language on a finite domain, the quantified constraint satisfaction problem is either in [Formula: see text] or PSpace-complete. Additionally, we build a constraint language on a 6-element domain such that the quantified constraint satisfaction problem over this language is [Formula: see text]-complete.

The Quantified Constraint Satisfaction Problem is the problem of evaluating a sentence with both quantifiers, over relations from some constraint language, with conjunction as the only connective. We show that for any constraint language on a finite domain the Quantified Constraint Satisfaction Problem is either in$\Pi_{2}^{P}$, or PSpace-complete. Additionally, we build a constraint language on a 6-element domain such that the Quantified Constraint Satisfaction Problem over this language is$\Pi_{2}^{P}$-complete.

We prove that QCSP(ℕ; x = y → y = z) is PSpace-complete, settling a question open for more than ten years. This completes the complexity classification for the QCSP over equality languages as a trichotomy between Logspace, NP-complete and PSpace-complete. We additionally settle the classification for bounded alternation QCSP(Γ), for Γ an equality language. Such problems are either in Logspace, NP-complete, co-NP-complete or rise in complexity in the Polynomial Hierarchy.

Let 𝔸 be an idempotent algebra on a finite domain. By mediating between results of Chen [ 1 ] and Zhuk [ 2 ], we argue that if 𝔸 satisfies the polynomially generated powers property (PGP) and ℬ is a constraint language invariant under 𝔸 (i.e., in Inv(𝔸)), then QCSP ℬ is in NP. In doing this, we study the special forms of PGP, switchability, and collapsibility, in detail, both algebraically and logically, addressing various questions such as decidability on the way. We then prove a complexity-theoretic converse in the case of infinite constraint languages encoded in propositional logic, that if Inv}(𝔸) satisfies the exponentially generated powers property (EGP), then QCSP (Inv(𝔸)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying what we term the Revised Chen Conjecture . This result becomes more significant now that the original Chen Conjecture (see [ 3 ]) is known to be false [ 4 ]. Switchability was introduced by Chen [ 1 ] as a generalization of the already-known collapsibility [ 5 ]. There, an algebra 𝔸 :=({ 0,1,2}; r ) was given that is switchable and not collapsible. We prove that, for all finite subsets Δ of Inv (𝔸 A), Pol (Δ) is collapsible. The significance of this is that, for QCSP on finite structures, it is still possible all QCSP tractability (in NP) explained by switchability is already explained by collapsibility. At least, no counterexample is known to this.

We give a surprising classification for the computational complexity of the Quantified Constraint Satisfaction Problem over a constraint language Γ, QCSP(Γ), where Γ is a finite language over three elements that contains all constants. In particular, such problems are in P, NP-complete, co-NP-complete, or PSpace-complete. Our classification refutes the hitherto widely believed Chen Conjecture. Additionally, we show that already on a 4-element domain there exists a constraint language Γ such that QCSP(Γ) is DP-complete (from Boolean Hierarchy), and on a 10-element domain there exists a constraint language giving the complexity class Θ P 2 . Meanwhile, we prove the Chen Conjecture for finite conservative languages Γ. If the polymorphism clone of such Γ has the polynomially generated powers property, then QCSP(Γ) is in NP. Otherwise, the polymorphism clone of Γ has the exponentially generated powers property and QCSP(Γ) is PSpace-complete. 1

For relational structures A, B of the same signature, the Promise Constraint Satisfaction Problem PCSP(A,B) asks whether a given input structure maps homomorphically to A or does not even map to B. We are promised that the input satisfies exactly one of these two cases. If there exists a structure C with homomorphisms $A\to C\to B$, then PCSP(A,B) reduces naturally to CSP(C). To the best of our knowledge all known tractable PCSPs reduce to tractable CSPs in this way. However Barto showed that some PCSPs over finite structures A, B require solving CSPs over infinite C. We show that even when such a reduction to finite C is possible, this structure may become arbitrarily large. For every integer $n>1$ and every prime p we give A, B of size n with a single relation of arity $n^p$ such that PCSP(A, B) reduces via a chain of homomorphisms $ A\to C\to B$ to a tractable CSP over some C of size p but not over any smaller structure. In a second family of examples, for every prime $p\geq 7$ we construct A, B of size $p-1$ with a single ternary relation such that PCSP(A, B) reduces via $A\to C\to B$ to a tractable CSP over some C of size p but not over any smaller structure. In contrast we show that if A, B are graphs and PCSP(A,B) reduces to tractable CSP(C) for some finite digraph C, then already A or B has a tractable CSP. This extends results and answers a question of Deng et al.

This paper focuses on the algebraic theory underlying the study of the complexity and the algorithms for the Constraint Satisfaction Problem (CSP). We unify, simplify, and extend parts of the three approaches that have been developed to study the CSP over finite templates – absorption theory that was used to characterize CSPs solvable by local consistency methods (JACM’14), and Bulatov’s and Zhuk’s theories that were used for two independent proofs of the CSP Dichotomy Theorem (FOCS’17, JACM’20).As the first contribution we present an elementary theorem about primitive positive definability and use it to obtain the starting points of Bulatov’s and Zhuk’s proofs as corollaries. As the second contribution we propose and initiate a systematic study of minimal Taylor algebras. This class of algebras is broad enough so that it suffices to verify the CSP Dichotomy Theorem on this class only, but still is unusually well behaved. In particular, many concepts from the three approaches coincide in the class, which is in striking contrast with the general setting.We believe that the theory initiated in this paper will eventually result in a simple and more natural proof of the Dichotomy Theorem that employs a simpler and more efficient algorithm, and will help in attacking complexity questions in other CSP-related problems.

The Surjective Constraint Satisfaction Problem (SCSP) is the problem of deciding whether there exists a surjective assignment to a set of variables subject to some specified constraints, where a surjective assignment is an assignment containing all elements of the domain. In this paper we show that the most famous SCSP, called No-Rainbow Problem, is NP-Hard. Additionally, we disprove the conjecture saying that the SCSP over a constraint language Γ and the CSP over the same language with constants have the same computational complexity up to poly-time reductions. Our counter-example also shows that the complexity of the SCSP cannot be described in terms of polymorphisms of the constraint language.

We give a surprising classification for the computational complexity of the Quantified Constraint Satisfaction Problem over a constraint language Γ, QCSP(Γ), where Γ is a finite language over 3 elements which contains all constants. In particular, such problems are either in P, NP-complete, co-NP-complete or PSpace-complete. Our classification refutes the hitherto widely-believed Chen Conjecture.

We consider finitary relations (also known as crosses) that are definable via finite disjunctions of unary relations, i.e. subsets, taken from a fixed finite parameter set Γ. We prove that whenever Γ contains at least one non-empty relation distinct from the full carrier set, there is a countably infinite number of polymorphism clones determined by relations that are disjunctively definable from Γ. Finally, we extend our result to finitely related polymorphism clones and countably infinite sets Γ. These results address an open problem raised in Creignou, N., et al. Theory Comput. Syst. 42 (2), 239–255 ( 2008 ), which is connected to the complexity analysis of the satisfiability problem of certain multiple-valued logics studied in Hähnle, R. Proc. 31st ISMVL 2001, 137–146 ( 2001 ).

Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. The standard way to parameterize interesting subclasses of the constraint satisfaction problem is via finite constraint languages. The main problem is to classify those subclasses that are solvable in polynomial time and those that are NP-complete. It was conjectured that if a constraint language has a weak near-unanimity polymorphism then the corresponding constraint satisfaction problem is tractable; otherwise, it is NP-complete. In the article, we present an algorithm that solves Constraint Satisfaction Problem in polynomial time for constraint languages having a weak near unanimity polymorphism, which proves the remaining part of the conjecture. 1

Let A be an idempotent algebra on a finite domain. We combine results of Chen, Zhuk and Carvalho et al. to argue that if A satisfies the polynomially generated powers property (PGP), then QCSP(Inv(A)) is in NP. We then use the result of Zhuk to prove a converse, that if Inv(A) satisfies the exponentially generated powers property (EGP), then QCSP(Inv(A)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying the moral correctness of what we term the Chen Conjecture. We examine in closer detail the situation for domains of size three. Over any finite domain, the only type of PGP that can occur is switchability. Switchability was introduced by Chen as a generalisation of the already-known Collapsibility. For three-element domain algebras A that are Switchable, we prove that for every finite subset Delta of Inv(A), Pol(Delta) is Collapsible. The significance of this is that, for QCSP on finite structures (over three-element domain), all QCSP tractability explained by Switchability is already explained by Collapsibility. Finally, we present a three-element domain complexity classification vignette, using known as well as derived results.