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Documents authored by Opršal, Jakub

Document
DOI: 10.4230/LIPIcs.ICDT.2024.10
Right-Adjoints for Datalog Programs

Authors: Balder ten Cate, Víctor Dalmau, and Jakub Opršal

Published in: LIPIcs, Volume 290, 27th International Conference on Database Theory (ICDT 2024)

Abstract
A Datalog program can be viewed as a syntactic specification of a mapping from database instances over some schema to database instances over another schema. We establish a large class of Datalog programs for which this mapping admits a (generalized) right-adjoint. We employ these results to obtain new insights into the existence of, and methods for constructing, homomorphism dualities within restricted classes of instances. From this, we derive new results regarding the existence of uniquely characterizing data examples for database queries in the presence of integrity constraints.
Cite as

Balder ten Cate, Víctor Dalmau, and Jakub Opršal. Right-Adjoints for Datalog Programs. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{tencate_et_al:LIPIcs.ICDT.2024.10,
  author =	{ten Cate, Balder and Dalmau, V{\'\i}ctor and Opr\v{s}al, Jakub},
  title =	{{Right-Adjoints for Datalog Programs}},
  booktitle =	{27th International Conference on Database Theory (ICDT 2024)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-312-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{290},
  editor =	{Cormode, Graham and Shekelyan, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2024.10},
  URN =		{urn:nbn:de:0030-drops-197929},
  doi =		{10.4230/LIPIcs.ICDT.2024.10},
  annote =	{Keywords: Datalog, Adjoints, Homomorphism Dualities, Database Constraints, Conjunctive Queries, Data Examples}
}
Document
DOI: 10.4230/LIPIcs.STACS.2024.34
Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs

Authors: Marek Filakovský, Tamio-Vesa Nakajima, Jakub Opršal, Gianluca Tasinato, and Uli Wagner

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Abstract
A linearly ordered (LO) k-colouring of a hypergraph is a colouring of its vertices with colours 1, … , k such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO k-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the "linearly ordered chromatic number" of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO 3-colourable, and the case that it is not even LO 4-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opršal, Wrochna, and Živný (2023).
Cite as

Marek Filakovský, Tamio-Vesa Nakajima, Jakub Opršal, Gianluca Tasinato, and Uli Wagner. Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 34:1-34:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{filakovsky_et_al:LIPIcs.STACS.2024.34,
  author =	{Filakovsk\'{y}, Marek and Nakajima, Tamio-Vesa and Opr\v{s}al, Jakub and Tasinato, Gianluca and Wagner, Uli},
  title =	{{Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{34:1--34:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.34},
  URN =		{urn:nbn:de:0030-drops-197445},
  doi =		{10.4230/LIPIcs.STACS.2024.34},
  annote =	{Keywords: constraint satisfaction problem, hypergraph colouring, promise problem, topological methods}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.34
Revisiting Alphabet Reduction in Dinur’s PCP

Authors: Venkatesan Guruswami, Jakub Opršal, and Sai Sandeep

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Abstract
Dinur’s celebrated proof of the PCP theorem alternates two main steps in several iterations: gap amplification to increase the soundness gap by a large constant factor (at the expense of much larger alphabet size), and a composition step that brings back the alphabet size to an absolute constant (at the expense of a fixed constant factor loss in the soundness gap). We note that the gap amplification can produce a Label Cover CSP. This allows us to reduce the alphabet size via a direct long-code based reduction from Label Cover to a Boolean CSP. Our composition step thus bypasses the concept of Assignment Testers from Dinur’s proof, and we believe it is more intuitive - it is just a gadget reduction. The analysis also uses only elementary facts (Parseval’s identity) about Fourier Transforms over the hypercube.
Cite as

Venkatesan Guruswami, Jakub Opršal, and Sai Sandeep. Revisiting Alphabet Reduction in Dinur’s PCP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{guruswami_et_al:LIPIcs.APPROX/RANDOM.2020.34,
  author =	{Guruswami, Venkatesan and Opr\v{s}al, Jakub and Sandeep, Sai},
  title =	{{Revisiting Alphabet Reduction in Dinur’s PCP}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{34:1--34:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.34},
  URN =		{urn:nbn:de:0030-drops-126372},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.34},
  annote =	{Keywords: PCP theorem, CSP, discrete Fourier analysis, label cover, long code}
}
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