2 Search Results for "Hladký, Jan"


Document
Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics

Authors: Sebastian Brandt, Yi-Jun Chang, Jan Grebík, Christoph Grunau, Václav Rozhoň, and Zoltán Vidnyánszky

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)


Abstract
We study connections between three different fields: distributed local algorithms, finitary factors of iid processes, and descriptive combinatorics. We focus on two central questions: Can we apply techniques from one of the areas to obtain results in another? Can we show that complexity classes coming from different areas contain precisely the same problems? We give an affirmative answer to both questions in the context of local problems on regular trees: 1) We extend the Borel determinacy technique of Marks [Marks - J. Am. Math. Soc. 2016] coming from descriptive combinatorics and adapt it to the area of distributed computing, thereby obtaining a more generally applicable lower bound technique in descriptive combinatorics and an entirely new lower bound technique for distributed algorithms. Using our new technique, we prove deterministic distributed Ω(log n)-round lower bounds for problems from a natural class of homomorphism problems. Interestingly, these lower bounds seem beyond the current reach of the powerful round elimination technique [Brandt - PODC 2019] responsible for all substantial locality lower bounds of the last years. Our key technical ingredient is a novel ID graph technique that we expect to be of independent interest; in fact, it has already played an important role in a new lower bound for the Lovász local lemma in the Local Computation Algorithms model from sequential computing [Brandt, Grunau, Rozhoň - PODC 2021]. 2) We prove that a local problem admits a Baire measurable coloring if and only if it admits a local algorithm with local complexity O(log n), extending the classification of Baire measurable colorings of Bernshteyn [Bernshteyn - personal communication]. A key ingredient of the proof is a new and simple characterization of local problems that can be solved in O(log n) rounds. We complement this result by showing separations between complexity classes from distributed computing, finitary factors, and descriptive combinatorics. Most notably, the class of problems that allow a distributed algorithm with sublogarithmic randomized local complexity is incomparable with the class of problems with a Borel solution. We hope that our treatment will help to view all three perspectives as part of a common theory of locality, in which we follow the insightful paper of [Bernshteyn - arXiv 2004.04905].

Cite as

Sebastian Brandt, Yi-Jun Chang, Jan Grebík, Christoph Grunau, Václav Rozhoň, and Zoltán Vidnyánszky. Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 29:1-29:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{brandt_et_al:LIPIcs.ITCS.2022.29,
  author =	{Brandt, Sebastian and Chang, Yi-Jun and Greb{\'\i}k, Jan and Grunau, Christoph and Rozho\v{n}, V\'{a}clav and Vidny\'{a}nszky, Zolt\'{a}n},
  title =	{{Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{29:1--29:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.29},
  URN =		{urn:nbn:de:0030-drops-156259},
  doi =		{10.4230/LIPIcs.ITCS.2022.29},
  annote =	{Keywords: Distributed Algorithms, Descriptive Combinatorics}
}
Document
An Approximate Version of the Tree Packing Conjecture via Random Embeddings

Authors: Julia Böttcher, Jan Hladký, Diana Piguet, and Anusch Taraz

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)


Abstract
We prove that for any pair of constants a>0 and D and for n sufficiently large, every family of trees of orders at most n, maximum degrees at most D, and with at most n(n-1)/2 edges in total packs into the complete graph of order (1+a)n. This implies asymptotic versions of the Tree Packing Conjecture of Gyarfas from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof.

Cite as

Julia Böttcher, Jan Hladký, Diana Piguet, and Anusch Taraz. An Approximate Version of the Tree Packing Conjecture via Random Embeddings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 490-499, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


Copy BibTex To Clipboard

@InProceedings{bottcher_et_al:LIPIcs.APPROX-RANDOM.2014.490,
  author =	{B\"{o}ttcher, Julia and Hladk\'{y}, Jan and Piguet, Diana and Taraz, Anusch},
  title =	{{An Approximate Version of the Tree Packing Conjecture via Random Embeddings}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{490--499},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.490},
  URN =		{urn:nbn:de:0030-drops-47184},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.490},
  annote =	{Keywords: tree packing conjecture, Ringel’s conjecture, random walks, quasirandom graphs}
}
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