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DOI: 10.4230/LIPIcs.STACS.2012.408

On the treewidth and related parameters of random geometric graphs

Authors: Dieter Mitsche and Guillem Perarnau

Published in: LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

Abstract

We give asymptotically exact values for the treewidth tw(G) of a random geometric graph G(n,r) in [0,sqrt(n)]^2. More precisely, we show that there exists some c_1 > 0, such that for any constant 0 < r < c_1, tw(G)=Theta(log(n)/loglog(n)), and also, there exists some c_2 > c_1, such that for any r=r(n)> c_2, tw(G)=Theta(r sqrt(n)). Our proofs show that for the corresponding values of r the same asymptotic bounds also hold for the pathwidth and treedepth of a random geometric graph.

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Dieter Mitsche and Guillem Perarnau. On the treewidth and related parameters of random geometric graphs. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 408-419, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{mitsche_et_al:LIPIcs.STACS.2012.408,
  author =	{Mitsche, Dieter and Perarnau, Guillem},
  title =	{{On the treewidth and related parameters of random geometric graphs}},
  booktitle =	{29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
  pages =	{408--419},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-35-4},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{14},
  editor =	{D\"{u}rr, Christoph and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.408},
  URN =		{urn:nbn:de:0030-drops-34280},
  doi =		{10.4230/LIPIcs.STACS.2012.408},
  annote =	{Keywords: Random geometric graphs, treewidth, treedepth}
}

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DOI: 10.4230/LIPIcs.FSTTCS.2008.1750

A new upper bound for 3-SAT

Authors: Josep Diaz, Lefteris Kirousis, Dieter Mitsche, and Xavier Perez-Gimenez

Published in: LIPIcs, Volume 2, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2008)

Abstract

We show that a randomly chosen $3$-CNF formula over $n$ variables with clauses-to-variables ratio at least $4.4898$ is asymptotically almost surely unsatisfiable. The previous best such bound, due to Dubois in 1999, was $4.506$. The first such bound, independently discovered by many groups of researchers since 1983, was $5.19$. Several decreasing values between $5.19$ and $4.506$ were published in the years between. The probabilistic techniques we use for the proof are, we believe, of independent interest.

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Josep Diaz, Lefteris Kirousis, Dieter Mitsche, and Xavier Perez-Gimenez. A new upper bound for 3-SAT. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 2, pp. 163-174, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{diaz_et_al:LIPIcs.FSTTCS.2008.1750,
  author =	{Diaz, Josep and Kirousis, Lefteris and Mitsche, Dieter and Perez-Gimenez, Xavier},
  title =	{{A new upper bound for 3-SAT}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{163--174},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-08-8},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{2},
  editor =	{Hariharan, Ramesh and Mukund, Madhavan and Vinay, V},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2008.1750},
  URN =		{urn:nbn:de:0030-drops-17507},
  doi =		{10.4230/LIPIcs.FSTTCS.2008.1750},
  annote =	{Keywords: Satisfiability, 3-SAT, random, threshold}
}

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On the treewidth and related parameters of random geometric graphs

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A new upper bound for 3-SAT

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