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Dmitriy Zhuk

Publications and source records attributed to Dmitriy Zhuk.

25 records · Page 2Linked to original sources
arXiv2019

A modification of the CSP algorithm for infinite languages

Constraint Satisfaction Problem on finite sets is known to be NP-complete in general but certain restrictions on the constraint language can ensure tractability. It was proved 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 a modification of the algorithm that works in polynomial time even for infinite constraint languages.

cs.CC↗
arXiv2018

The number of clones determined by disjunctions of unary relations

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 $Γ$.

math.LO↗
arXiv2017

The complexity of quantified constraints using the algebraic formulation

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 QCSP(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.

cs.LO↗
arXiv2016

Key (critical) relations preserved by a weak near-unanimity function

In the paper we introduce a notion of a key relation, which is similar to the notion of a critical relation introduced by Keith A.Kearnes and Ágnes Szendrei. All clones on finite sets can be defined by only key relations. In addition there is a nice description of all key relations on 2 elements. These are exactly the relations that can be defined as a disjunction of linear equations. In the paper we show that, in general key relations do not have such a nice description. Nevertheless, we obtain a nice characterization of all key relations preserved by a weak near-unanimity function. This characterization is presented in the paper.

math.RA↗
arXiv2015

Switchability and collapsibility of Gap Algebras

Let A be an idempotent algebra on a 3-element domain D that omits a G-set for a factor. Suppose A is not αβ-projective (for some alpha, beta subsets of D) and is not collapsible. It follows that A is switchable. We prove that, for every finite subset Delta of Inv(A), Pol(Delta) is collapsible. We also exhibit an algebra that is collapsible from a non-singleton source but is not collapsible from any singleton source.

cs.LO↗
arXiv2015

The Size of Generating Sets of Powers

In the paper we prove for every finite algebra A that either it has the polynomially generated powers (PGP) property, or it has the exponentially generated powers (EGP) property. For idempotent algebras we give a simple criteria for the algebra to satisfy EGP property.

math.RA↗