We prove that any perfect quantum strategy for the two-prover game encoding a constraint satisfaction problem (CSP) can be simulated via a perfect classical strategy with an extra classical communication channel, whose size depends only on (i) the size of the shared quantum system used in the quantum strategy, and (ii) structural parameters of the CSP template. The result is obtained via a combinatorial characterisation of perfect classical strategies with extra communication channels and a geometric rounding procedure for the projection-valued measurements involved in quantum strategies.A key intermediate step of our proof is to establish that the gap between the classical chromatic number of graphs and its quantum variant is bounded when the quantum strategy involves shared quantum information of bounded size.

Two major milestones on the road to the full complexity dichotomy for finite-domain constraint satisfaction problems were Bulatov’s proof of the dichotomy for conservative templates, and the structural dichotomy for smooth digraphs of algebraic length 1 due to Barto, Kozik, and Niven. We lift the combined scenario to the infinite, and prove that any smooth digraph of algebraic length 1 pp-constructs, together with pairs of orbits of an oligomorphic subgroup of its automorphism group, every finite structure – and hence its conservative graph-colouring problem is NP-hard – unless the digraph has a pseudo-loop, i.e. an edge within an orbit. We thereby overcome, for the first time, previous obstacles to lifting structural results for digraphs in this context from finite to ω-categorical structures; the strongest lifting results hitherto not going beyond a genera-lisation of the Hell-Nešetřil theorem for undirected graphs. As a consequence, we obtain a new algebraic invariant of arbitrary ω-categorical structures enriched by pairs of orbits which fail to pp-construct some finite structure.

The 1 -in- 3 and N ot -A ll -E qual satisfiability problems for Boolean CNF formulas are two well-known NP -hard problems. In contrast, the promise 1 -in- 3 vs . N ot -A ll -E qual problem can be solved in polynomial time. In the present work, we investigate this constraint satisfaction problem in a regime where the promise is weakened from either side by a rainbow-free structure and establish a complexity dichotomy for the resulting class of computational problems.

We present a template for the Promise Constraint Satisfaction Problem (PCSP) which is NP-hard but does not satisfy the current state-of-the-art hardness condition [ACMTCT'21]. We introduce a new "injective" condition based on the smooth version of the layered PCP Theorem and use this new condition to confirm that the problem is indeed NP-hard.

The 1-in-3 and Not-All-Eqal satisfiability problems for Boolean CNF formulas are two well-known NP-hard problems. In contrast, the promise 1-in-3 vs. Not-All-Eqal problem can be solved in polynomial time. In the present work, we investigate this constraint satisfaction problem in a regime where the promise is weakened from either side by a rainbow-free structure, and establish a complexity dichotomy for the resulting class of computational problems.

We investigate structural implications arising from the condition that a given directed graph does not interpret, in the sense of primitive positive interpretation with parameters or orbits, every finite structure. Our results generalize several theorems from the literature and yield further algebraic invariance properties that must be satisfied in every such graph. Algebraic properties of this kind are tightly connected to the tractability of constraint satisfaction problems, and we obtain new such properties even for infinite countably categorical graphs. We balance these positive results by showing the existence of a countably categorical hypergraph that fails to interpret some finite structure, while still lacking some of the most essential algebraic invariance properties known to hold for finite structures.

A value of a CSP instance is typically defined as a fraction of constraints that can be simultaneously met. We propose an alternative definition of a value of an instance and show that, for purely combinatorial reasons, a value of an unsolvable instance is bounded away from one; we call this fact a gap theorem. We show that the gap theorem implies NP-hardness of a gap version of the Layered Label Cover Problem. The same result can be derived from the PCP Theorem, but a full, self-contained proof of our reduction is quite short and the result can still provide PCP–free NP–hardness proofs for numerous problems. The simplicity of our reasoning also suggests that weaker versions of Unique-Games-type conjectures, e.g., the d-to-1 conjecture, might be accessible and serve as an intermediate step for proving these conjectures in their full strength. As the second, main application we provide a sufficient condition under which a fixed template Promise Constraint Satisfaction Problem (PCSP) reduces to another PCSP. The correctness of the reduction hinges on the gap theorem, but the reduction itself is very simple. As a consequence, we obtain that every CSP can be canonically reduced to most of the known NP-hard PCSPs, such as the approximate hypergraph coloring problem.

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 characterization of all the constraint satisfaction problems solvable by local consistency checking (also known as CSPs of bounded width) was proposed by Feder and Vardi [ SIAM J. Comput., 28 (1998), pp. 57--104]. It was confirmed by two independent proofs by Bulatov [ Bounded Relational Width, manuscript, 2009] and Barto and Kozik [L. Barto and M. Kozik, 50th Annual IEEE Symposium on Foundations of Computer Science, 2009, pp. 595--603], [L. Barto and M. Kozik, J. ACM, 61 (2014), 3]. Later Barto [ J. Logic Comput., 26 (2014), pp. 923--943] proved a collapse of the hierarchy of local consistency notions by showing that (2,3) minimality solves all the CSPs of bounded width. In this paper we present a new consistency notion, jpq consistency, which also solves all the CSPs of bounded width. Our notion is strictly weaker than (2,3) consistency, (2,3) minimality, path consistency, and singleton arc consistency (SAC). This last fact allows us to answer the question of Chen, Dalmau, and Grußien [ J. Logic Comput., 23 (2013), pp. 87--108] by confirming that SAC solves all the CSPs of bounded width. Moreover, as known algorithms work faster for SAC, the result implies that CSPs of bounded width can be, in practice, solved more efficiently. The definition of jpq consistency is closely related to a consistency condition obtained from the rounding of an SDP relaxation of a CSP instance. In fact, the main result of this paper is used by Dalmau et al. [ Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, ACM, New York, 2017, pp. 340--357] to show that CSPs with near unanimity polymorphisms admit robust approximation algorithms with polynomial loss. Finally, an algebraic characterization of some term conditions satisfied in algebras associated with templates of bounded width, first proved by Brady, is a direct consequence of our result.

A PCSP is a combination of two CSPs defined by two similar templates; the computational question is to distinguish a YES instance of the first one from a NO instance of the second. The computational complexity of many PCSPs remains unknown. Even the case of Boolean templates (solved for CSP by Schaefer [STOC'78]) remains wide open. The main result of Brakensiek and Guruswami [SODA'18] shows that Boolean PCSPs exhibit a dichotomy (PTIME vs. NPC) when "all the clauses are symmetric and allow for negation of variables''. In this paper we remove the "allow for negation of variables'' assumption from the theorem. The "symmetric" assumption means that changing the order of variables in a constraint does not change its satisfiability. The "negation of variables" means that both of the templates share a relation which can be used to effectively negate Boolean variables. The main result of this paper establishes dichotomy for all the symmetric boolean templates. The tractability case of our theorem and the theorem of Brakensiek and Guruswami are almost identical. The main difference, and the main contribution of this work, is the new reason for hardness and the reasoning proving the split.

An instance of the constraint satisfaction problem (CSP) is given by a family of constraints on overlapping sets of variables, and the goal is to assign values from a fixed domain to the variables so that all constraints are satisfied. In the optimization version, the goal is to maximize the number of satisfied constraints. An approximation algorithm for a CSP is called robust if it outputs an assignment satisfying an $(1-g(\varepsilon))$-fraction of constraints on any $(1-\varepsilon)$-satisfiable instance, where the loss function $g$ is such that $g(\varepsilon)\rightarrow 0$ as $\varepsilon\rightarrow 0$. We study how the robust approximability of CSPs depends on the set of constraint relations allowed in instances, the so-called constraint language. All constraint languages admitting a robust polynomial-time algorithm (with some $g$) have been characterized by Barto and Kozik, with the general bound on the loss $g$ being doubly exponential, specifically $g(\varepsilon)=O((\log\log(1/\varepsilon))/\log(1/\varepsilon))$. It is natural to ask when a better loss can be achieved, in particular polynomial loss $g(\varepsilon)=O(\varepsilon^{1/k})$ for some constant $k$. In this paper, we consider CSPs with a constraint language having a near-unanimity polymorphism. This general condition almost matches a known necessary condition for having a robust algorithm with polynomial loss. We give two randomized robust algorithms with polynomial loss for such CSPs: one works for any near-unanimity polymorphism and the parameter $k$ in the loss depends on the size of the domain and the arity of the relations in $\Gamma$, while the other works for a special ternary near-unanimity operation called the dual discriminator with $k=2$ for any domain size. In the latter case, the CSP is a common generalization of Unique Games with a fixed domain and 2-Sat. In the former case, we use the algebraic approach to the CSP. Both cases use the standard semidefinite programming relaxation for the CSP.

An instance of the Constraint Satisfaction Problem (CSP) is given by a family of constraints on overlapping sets of variables, and the goal is to assign values from a fixed domain to the variables so that all constraints are satisfied. In the optimization version, the goal is to maximize the number of satisfied constraints. An approximation algorithm for CSP is called robust if it outputs an assignment satisfying a (1 — g(∊))-fraction of constraints on any (1 — ∊)-satisfiable instance, where the loss function g is such that g(∊) → 0 as ∊ → 0. We study how the robust approximability of CSPs depends on the set of constraint relations allowed in instances, the so-called constraint language. All constraint languages admitting a robust polynomial-time algorithm (with some g) have been characterised by Barto and Kozik, with the general bound on the loss g being doubly exponential, specifically g(∊) = O((loglog(1/ ∊))/log(1/ ∊)). It is natural to ask when a better loss can be achieved: in particular, polynomial loss g(∊) = O(∊1/k) for some constant k. In this paper, we consider CSPs with a constraint language having a near- unanimity polymorphism. We give two randomized robust algorithms with polynomial loss for such CSPs: one works for any near-unanimity polymorphism and the parameter k in the loss depends on the size of the domain and the arity of the relations in Γ, while the other works for a special ternary near-unanimity operation called dual discriminator with k = 2 for any domain size. In the latter case, the CSP is a common generalisation of Unique Games with a fixed domain and 2-Sat. In the former case, we use the algebraic approach to the CSP. Both cases use the standard semidefinite programming relaxation for CSP.

The characterization of all the Constraint Satisfaction Problems of bounded width, proposed by Feder and Vardi [SICOMP'98], was confirmed in [Bulatov'09] and independently in [FOCS'09, JACM'14]. Both proofs are based on the (2,3)-consistency (using Prague consistency in [FOCS'09], directly in [Bulatov'09]) which is costly to verify.

An algorithm for a constraint satisfaction problem is called robust if it outputs an assignment satisfying at least $(1-g(\varepsilon))$-fraction of the constraints given a $(1-\varepsilon)$-satisfiable instance, where $g(\varepsilon) \rightarrow 0$ as $\varepsilon \rightarrow 0$. Guruswami and Zhou conjectured a characterization of constraint languages for which the corresponding constraint satisfaction problem admits an efficient robust algorithm. This paper confirms their conjecture.

We prove that constraint satisfaction problems without the ability to count are solvable by the local consistency checking algorithm. This settles three (equivalent) conjectures: Feder--Vardi [SICOMP’98], Bulatov [LICS’04] and Larose--Zádori [AU’07].

We show that if a finite relational structure has a near unanimity polymorphism, then the constraint satisfaction problem with that structure as its fixed template has bounded pathwidth duality, putting the problem in nondeterministic logspace. This generalizes the analogous result of Dalmau and Krokhin for majority polymorphisms and lends further support to a conjecture suggested by Larose and Tesson.

An algorithm for a constraint satisfaction problem is called robust if it outputs an assignment satisfying at least (1-g(ε))-fraction of the constraints given a (1-ε)-satisfiable instance, where g(ε) -> 0 as ε -> 0, $g(0)=0. Guruswami and Zhou conjectured a characterization of constraint languages for which the corresponding constraint satisfaction problem admits an efficient robust algorithm. This paper confirms their conjecture.

We provide two new characterizations for finitely generated varieties with Taylor terms. The first characterization is using "absorbing sets" and the second one "cyclic operations". These new conditions allow us to reprove the conjecture of Bang-Jensen and Hell (proved by the authors, comp. STOC'08, SICOMP'09) and the characterization of locally finite Taylor varieties using weak near-unanimity operations (proved by McKenzie and Maroti, Alg.Univ. 2009) in an elementary and self-contained way. The research is closely connected to the algebraic approach to CSP and previous results obtained by authors using similar tools [comp. STOC'08, SICOMP'09, FOCS'09 etc.].

We provide a full characterization of applicability of The Local Consistency Checking algorithm to solving the non-uniform Constraint Satisfaction Problems. This settles the conjecture of Larose and Zadori.

We show that a constraint language with compatible Jónsson terms (or, equivalently, associated with an algebra generating a congruence distributive variety) defines a constraint satisfaction problem solvable by the local consistency checking algorithm.

Bang-Jensen and Hell conjectured in 1990 (using the language of graph homomorphisms) a constraint satisfaction problem (CSP) dichotomy for digraphs with no sources or sinks. The conjecture states that the CSP for such a digraph is tractable if each component of its core is a cycle and is $NP$-complete otherwise. In this paper we prove this conjecture and, as a consequence, a conjecture of Bang-Jensen, Hell, and MacGillivray from 1995 classifying hereditarily hard digraphs. Further, we show that the CSP dichotomy for digraphs with no sources or sinks agrees with the algebraic characterization conjectured by Bulatov, Jeavons, and Krokhin in 2005.

We construct a finite algebra generating a variety with 2EXPTIME complete membership problem. This proves that the universal membership problem for varieties and the varietal equivalence problem are 2EXPTIME complete as well, answering the question of Bergman and Slutzki from 2000.

We use a connection between polymorphisms and the structure of smooth digraphs to prove the conjecture of Bang-Jensen and Hell from 1990 and, as a consequence, a conjecture of Bang-Jensen, Hell and MacGillivray from 1995. The conjectured characterization of computationally complex coloring problems for smooth digraphs is proved using tools of universal algebra. We cite further graph results obtained using this new approach. The proofs are based in an universal algebraic framework developed for the Constraint Satisfaction Problem and the CSP dichotomy conjecture of Feder and Vardi in particular.