arXiv subjects

Dmitriy Zhuk

Publications and source records attributed to Dmitriy Zhuk.

At least 19 recordsLinked to original sources

arXiv2026

Toward a Uniform Algorithm and Uniform Reduction for Constraint Problems

We develop a unified framework to characterize the power of higher-level algorithms for the constraint satisfaction problem (CSP), such as $k$-consistency, the Sherali-Adams LP hierarchy, and the affine IP hierarchy. As a result, solvability of a fixed-template CSP or, more generally, a Promise CSP by a given level is shown to depend only on the polymorphism minion of the template. Similarly, we obtain a minion-theoretic description of $k$-consistency reductions between Promise CSPs. We introduce a new hierarchy of SDP-like vector relaxations with vectors over $\mathbb Z_{p}$ in which orthogonality is imposed on $k$-tuples of vectors. Surprisingly, this relaxation turns out to be equivalent to the $k$-th level of the AIP-$\mathbb{Z}_p$ relaxation. We show that it solves the CSP of the dihedral group $\mathbf{D}_4$, the smallest CSP that fools the singleton BLP+AIP algorithm. Using this vector representation, we further show that the $p$-th level of the $\mathbb{Z}_p$ relaxation solves linear equations modulo $p^2$.

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arXiv2025

Singleton algorithms for the Constraint Satisfaction Problem

A natural strengthening of an algorithm for the (promise) constraint satisfaction problem is its singleton version: we first fix a variable to an element from its domain, then run the algorithm, and remove the element from the domain if the answer is negative. Using the Hales-Jewett theorem, we characterize the power of the singleton versions of standard universal algorithms for the (promise) CSP over a fixed template in terms of the existence of polymorphisms with certain symmetries, which we call palette symmetric polymorphisms. By proving the existence of such polymorphisms we establish that the singleton version of the BLP+AIP algorithm solves all (multi-sorted) tractable CSPs over domains of size at most 7. We further show that already for domain size 8 there exists a relational structure arising from the dihedral group $\mathbf D_4$ that does not admit palette symmetric polymorphisms and cannot be solved by singleton BLP+AIP. By providing concrete CSP templates, we illustrate the limitations of linear programming, the power of the singleton versions, and the elegance of palette symmetric polymorphisms. Among tractable temporal templates, we exhibit a structure demonstrating that finiteness is crucial for the Hales-Jewett argument; nevertheless, by introducing generalized palette polymorphisms we establish tractability for each such template.

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arXiv2025

On the lattice of multi-sorted relational clones on a two-element set

We introduce a new approach to the description of multi-sorted clones (sets of $k$-tuples of operations of the same arity, closed under coordinatewise composition and containing all projection tuples) on a two-element domain. Leveraging the well-known Galois connection between operations and relations, we define a small class of canonical relations sufficient to describe all Boolean multi-sorted clones up to non-surjective operations. Furthermore, we introduce elementary operations on relations, which are less cumbersome than general formulas and have many useful properties. Using these tools, we provide a new and elementary proof of the famous Post's lattice theorem. We also show that every multi-sorted clone of $k$-tuples of operations decomposes into a surjective part described by canonical relations and $2k$ clones of $(k-1)$-tuples of operations. This structural understanding allows us to describe an embedding of the lattice of multi-sorted clones into a well-understood poset. In particular, we rederive - by a simpler method - a result of V. Taimanov originally from 1983, showing that every multi-sorted clone on a two-element domain is finitely generated. Finally, we also give a concise proof of the Galois connection between (surjective) multi-sorted clones and the corresponding closed sets of relations.

math.LO↗

arXiv2024

$\Pi_{2}^{P}$ vs PSpace Dichotomy for the Quantified Constraint Satisfaction Problem

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.

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arXiv2024

A simplified proof of the CSP Dichotomy Conjecture and XY-symmetric operations

We develop a new theory of strong subalgebras and linear congruences that are defined globally. Using this theory we provide a new proof of the correctness of Zhuk's algorithm for all tractable CSPs on a finite domain, and therefore a new simplified proof of the CSP Dichotomy Conjecture. Additionally, using the new theory we prove that composing a weak near-unanimity operation of an odd arity $n$ we can derive an $n$-ary operation that is symmetric on all two-element sets. Thus, CSP over a constraint language $\Gamma$ on a finite domain is tractable if and only if there exist infinitely many polymorphisms of $\Gamma$ that are symmetric on all two-element sets.

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arXiv2023

Submaximal clones over a three-element set up to minor-equivalence

We study clones modulo minor homomorphisms, which are mappings from one clone to another preserving arities of operations and respecting permutation and identification of variables. Minor-equivalent clones satisfy the same sets of identities of the form $f(x_1,\dots,x_n)\approx g(y_1,\dots,y_m)$, also known as minor identities, and therefore share many algebraic properties. Moreover, it was proved that the complexity of the $\operatorname{CSP}$ of a finite structure $\mathbb{A}$ only depends on the set of minor identities satisfied by the polymorphism clone of $\mathbb{A}$. In this article we consider the poset that arises by considering all clones over a three-element set with the following order: we write $\mathcal{C} \preceq_{\mathrm{m}} \mathcal{D}$ if there exists a minor homomorphism from $\mathcal{C}$ to $\mathcal{D}$. We show that the aforementioned poset has only three submaximal elements.

math.RA↗

arXiv2021

The complexity of the Quantified CSP having the polynomially generated powers property

It is known that if an algebra of polymorphisms of the constraint language has the Polynomially Generated Powers (PGP) Property then the Quantified CSP can be reduced to the CSP over the same constraint language with constants. The only limitation of this reduction is that it is applicable only for the constraint languages with constants. We drastically simplified the reduction and generalized it for constraint languages without constants. As a result, we completely classified the complexity of the QCSP for constraint languages having the PGP property.

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arXiv2021

Small Promise CSPs that reduce to large CSPs

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.

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arXiv2021

The lattice of clones of self-dual operations collapsed

There are continuum many clones on a three-element set even if they are considered up to \emph{homomorphic equivalence}. The clones we use to prove this fact are clones consisting of \emph{self-dual operations}, i.e., operations that preserve the relation $\{(0,1),(1,2),(2,0)\}$. However, there are only countably many such clones when considered up to equivalence with respect to \emph{minor-preserving maps} instead of clone homomorphisms. We give a full description of the set of clones of self-dual operations, ordered by the existence of minor-preserving maps. Our result can also be phrased as a statement about structures on a three-element set, ordered by primitive positive constructability, because there is a minor-preserving map from the polymorphism clone of a finite structure $\mathfrak A$ to the polymorphism clone of a finite structure $\mathfrak B$ if and only if there is a primitive positive construction of $\mathfrak B$ in $\mathfrak A$.

math.RA↗

arXiv2021

The complexity of quantified constraints: collapsibility, switchability and the algebraic formulation

Let A be an idempotent algebra on a finite domain. By mediating between results of Chen and Zhuk, we argue that if A satisfies the polynomially generated powers property (PGP) and B is a constraint language invariant under A (that is, 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 the original Chen Conjecture is known to be false. Switchability was introduced by Chen as a generalisation of the already-known collapsibility. For three-element domain algebras A that are switchable and omit a G-set, we prove that, for every finite subset D of Inv(A), Pol(D) is collapsible. The significance of this is that, for QCSP on finite structures (over a three-element domain), all QCSP tractability (in P) explained by switchability is already explained by collapsibility.

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arXiv2021

No-Rainbow Problem and the Surjective Constraint Satisfaction Problem

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.

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arXiv2021

Unifying the Three Algebraic Approaches to the CSP via Minimal Taylor Algebras

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 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 this 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.

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arXiv2021

The complete classification for quantified equality constraints

We prove that QCSP$(\mathbb{N};x=y\rightarrow y=z)$ 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$(\Gamma)$, for $\Gamma$ an equality language. Such problems are either in Logspace, NP-complete, co-NP-complete or rise in complexity in the Polynomial Hierarchy.

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arXiv2020

A Proof of the CSP Dichotomy Conjecture

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 paper 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.

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arXiv2020

Existence of cube terms in finite algebras

We study the problem of whether a given finite algebra with finitely many basic operations contains a cube term; we give both structural and algorithmic results. We show that if such an algebra has a cube term then it has a cube term of dimension at most $N$, where the number $N$ depends on the arities of basic operations of the algebra and the size of the basic set. For finite idempotent algebras we give a tight bound on $N$ that, in the special case of algebras with more than $\binom{|A|}2$ basic operations, improves an earlier result of K. Kearnes and A. Szendrei. On the algorithmic side, we show that deciding the existence of cube terms is in P for idempotent algebras and in EXPTIME in general. Since an algebra contains a $k$-ary near unanimity operation if and only if it contains a $k$-dimensional cube term and generates a congruence distributive variety, our algorithm also lets us decide whether a given finite algebra has a near unanimity operation.

math.RA↗

arXiv2020

Strong subalgebras and the Constraint Satisfaction Problem

In 2007 it was conjectured that the Constraint Satisfaction Problem (CSP) over a constraint language $Γ$ is tractable if and only if $Γ$ is preserved by a weak near-unanimity (WNU) operation. After many efforts and partial results, this conjecture was independently proved by Andrei Bulatov and the author in 2017. In this paper we consider one of two main ingredients of my proof, that is, strong subalgebras that allow us to reduce domains of the variables iteratively. To explain how this idea works we show the algebraic properties of strong subalgebras and provide self-contained proof of two important facts about the complexity of the CSP. First, we prove that if a constraint language is not preserved by a WNU operation then the corresponding CSP is NP-hard. Second, we characterize all constraint languages that can be solved by local consistency checking. Additionally, we characterize all idempotent algebras not having a WNU term of a concrete arity $n$, not having a WNU term, having WNU terms of all arities greater than 2. Most of the results presented in the paper are not new, but I believe this paper can help to understand my approach to CSP and the new self-contained proof of known facts will be also useful.

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arXiv2019

Deciding the existence of minority terms

This paper investigates the computational complexity of deciding if a given finite idempotent algebra has a ternary term operation $m$ that satisfies the minority equations $m(y,x,x) \approx m(x,y,x) \approx m(x,x,y) \approx y$. We show that a common polynomial-time approach to testing for this type of condition will not work in this case and that this decision problem lies in the class NP.

math.LO↗

arXiv2019

QCSP monsters and the demise of the Chen Conjecture

We give a surprising classification for the computational complexity of the Quantified Constraint Satisfaction Problem over a constraint language $\Gamma$, QCSP$(\Gamma)$, where $\Gamma$ 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. Additionally, we show that already on a 4-element domain there exists a constraint language $\Gamma$ such that QCSP$(\Gamma)$ is DP-complete (from Boolean Hierarchy), and on a 10-element domain there exists a constraint language giving the complexity class $\Theta_{2}^{P}$. Meanwhile, we prove the Chen Conjecture for finite conservative languages $\Gamma$. If the polymorphism clone of $\Gamma$ has the polynomially generated powers (PGP) property then QCSP$(\Gamma)$ is in NP. Otherwise, the polymorphism clone of $\Gamma$ has the exponentially generated powers (EGP) property and QCSP$(\Gamma)$ is PSpace-complete.

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