4 Search Results for "Zhuk, Dmitriy"

Document
DOI: 10.4230/LIPIcs.MFCS.2023.28
Short Definitions in Constraint Languages

Authors: Jakub Bulín and Michael Kompatscher

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract
A first-order formula is called primitive positive (pp) if it only admits the use of existential quantifiers and conjunction. Pp-formulas are a central concept in (fixed-template) constraint satisfaction since CSP(Γ) can be viewed as the problem of deciding the primitive positive theory of Γ, and pp-definability captures gadget reductions between CSPs. An important class of tractable constraint languages Γ is characterized by having few subpowers, that is, the number of n-ary relations pp-definable from Γ is bounded by 2^p(n) for some polynomial p(n). In this paper we study a restriction of this property, stating that every pp-definable relation is definable by a pp-formula of polynomial length. We conjecture that the existence of such short definitions is actually equivalent to Γ having few subpowers, and verify this conjecture for a large subclass that, in particular, includes all constraint languages on three-element domains. We furthermore discuss how our conjecture imposes an upper complexity bound of co-NP on the subpower membership problem of algebras with few subpowers.
Cite as

Jakub Bulín and Michael Kompatscher. Short Definitions in Constraint Languages. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bulin_et_al:LIPIcs.MFCS.2023.28,
  author =	{Bul{\'\i}n, Jakub and Kompatscher, Michael},
  title =	{{Short Definitions in Constraint Languages}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{28:1--28:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.28},
  URN =		{urn:nbn:de:0030-drops-185629},
  doi =		{10.4230/LIPIcs.MFCS.2023.28},
  annote =	{Keywords: constraint satisfaction, primitive positive definability, few subpowers, polynomially expressive, relational clone, subpower membership}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.16
Ideal Membership Problem for Boolean Minority and Dual Discriminator

Authors: Arpitha P. Bharathi and Monaldo Mastrolilli

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract
The polynomial Ideal Membership Problem (IMP) tests if an input polynomial f ∈ 𝔽[x_1,… ,x_n] with coefficients from a field 𝔽 belongs to a given ideal I ⊆ 𝔽[x_1,… ,x_n]. It is a well-known fundamental problem with many important applications, though notoriously intractable in the general case. In this paper we consider the IMP for polynomial ideals encoding combinatorial problems and where the input polynomial f has degree at most d = O(1) (we call this problem IMP_d). A dichotomy result between "hard" (NP-hard) and "easy" (polynomial time) IMPs was achieved for Constraint Satisfaction Problems over finite domains [Andrei A. Bulatov, 2017; Dmitriy Zhuk, 2020] (this is equivalent to IMP_0) and IMP_d for the Boolean domain [Mastrolilli, 2019], both based on the classification of the IMP through functions called polymorphisms. For the latter result, there are only six polymorphisms to be studied in order to achieve a full dichotomy result for the IMP_d. The complexity of the IMP_d for five of these polymorphisms has been solved in [Mastrolilli, 2019] whereas for the ternary minority polymorphism it was incorrectly declared in [Mastrolilli, 2019] to have been resolved by a previous result. In this paper we provide the missing link by proving that the IMP_d for Boolean combinatorial ideals whose constraints are closed under the minority polymorphism can be solved in polynomial time. This completes the identification of the precise borderline of tractability for the IMP_d for constrained problems over the Boolean domain. We also prove that the proof of membership for the IMP_d for problems constrained by the dual discriminator polymorphism over any finite domain can also be found in polynomial time. Bulatov and Rafiey [Andrei A. Bulatov and Akbar Rafiey, 2020] recently proved that the IMP_d for this polymorphism is decidable in polynomial time, without needing a proof of membership. Our result gives a proof of membership and can be used in applications such as Nullstellensatz and Sum-of-Squares proofs.
Cite as

Arpitha P. Bharathi and Monaldo Mastrolilli. Ideal Membership Problem for Boolean Minority and Dual Discriminator. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 16:1-16:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bharathi_et_al:LIPIcs.MFCS.2021.16,
  author =	{Bharathi, Arpitha P. and Mastrolilli, Monaldo},
  title =	{{Ideal Membership Problem for Boolean Minority and Dual Discriminator}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{16:1--16:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.16},
  URN =		{urn:nbn:de:0030-drops-144560},
  doi =		{10.4230/LIPIcs.MFCS.2021.16},
  annote =	{Keywords: Polynomial ideal membership, Polymorphisms, Gr\"{o}bner basis theory, Constraint satisfaction problems}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.13
Ideal Membership Problem and a Majority Polymorphism over the Ternary Domain

Authors: Arpitha P. Bharathi and Monaldo Mastrolilli

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract
The Ideal Membership Problem (IMP) asks if an input polynomial f ∈ 𝔽[x₁,… ,x_n] with coefficients from a field 𝔽 belongs to an input ideal I ⊆ 𝔽[x₁,… ,x_n]. It is a well-known fundamental problem with many important applications, though notoriously intractable in the general case. In this paper we consider the IMP for polynomial ideals encoding combinatorial problems and where the input polynomial f has degree at most d = O(1) (we call this problem IMP_d). Our main interest is in understanding when the inherent combinatorial structure of the ideals makes the IMP_d "hard" (NP-hard) or "easy" (polynomial time) to solve. Such a dichotomy result between "hard" and "easy" IMPs was recently achieved for Constraint Satisfaction Problems over finite domains [Andrei A. Bulatov, 2017; Dmitriy Zhuk, 2017] (this is equivalent to IMP₀) and IMP_d for the Boolean domain [Mastrolilli, 2019], both based on the classification of the IMP through functions called polymorphisms. For the latter result, each polymorphism determined the complexity of the computation of a suitable Gröbner basis. In this paper we consider a 3-element domain and a majority polymorphism (constraints under this polymorphism are a generalisation of the 2-SAT problem). By using properties of the majority polymorphism and assuming graded lexicographic ordering of monomials, we show that the reduced Gröbner basis of ideals whose varieties are closed under the majority polymorphism can be computed in polynomial time. This proves polynomial time solvability of the IMP_d for these constrained problems. We conjecture that this result can be extended to a general finite domain of size k = O(1). This is a first step towards the long term and challenging goal of generalizing the dichotomy results of solvability of the IMP_d for a finite domain.
Cite as

Arpitha P. Bharathi and Monaldo Mastrolilli. Ideal Membership Problem and a Majority Polymorphism over the Ternary Domain. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bharathi_et_al:LIPIcs.MFCS.2020.13,
  author =	{Bharathi, Arpitha P. and Mastrolilli, Monaldo},
  title =	{{Ideal Membership Problem and a Majority Polymorphism over the Ternary Domain}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{13:1--13:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.13},
  URN =		{urn:nbn:de:0030-drops-126829},
  doi =		{10.4230/LIPIcs.MFCS.2020.13},
  annote =	{Keywords: Polynomial ideal membership, Polymorphisms, Gr\"{o}bner basis theory, Constraint satisfaction problems}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2017.27
The Complexity of Quantified Constraints Using the Algebraic Formulation

Authors: Catarina Carvalho, Barnaby Martin, and Dmitriy Zhuk

Published in: LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Abstract
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.
Cite as

Catarina Carvalho, Barnaby Martin, and Dmitriy Zhuk. The Complexity of Quantified Constraints Using the Algebraic Formulation. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{carvalho_et_al:LIPIcs.MFCS.2017.27,
  author =	{Carvalho, Catarina and Martin, Barnaby and Zhuk, Dmitriy},
  title =	{{The Complexity of Quantified Constraints Using the Algebraic Formulation}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{27:1--27:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.27},
  URN =		{urn:nbn:de:0030-drops-80793},
  doi =		{10.4230/LIPIcs.MFCS.2017.27},
  annote =	{Keywords: Quantified Constraints, Computational Complexity, Universal Algebra, Constraint Satisfaction}
}
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