License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2020.50
URN: urn:nbn:de:0030-drops-127161
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12716/
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Jaffke, Lars ; de Oliveira Oliveira, Mateus ; Tiwary, Hans Raj

Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach

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Abstract

It can be shown that each permutation group G βŠ‘ π•Š_n can be embedded, in a well defined sense, in a connected graph with O(n+|G|) vertices. Some groups, however, require much fewer vertices. For instance, π•Š_n itself can be embedded in the n-clique K_n, a connected graph with n vertices. In this work, we show that the minimum size of a context-free grammar generating a finite permutation group GβŠ‘ π•Š_n can be upper bounded by three structural parameters of connected graphs embedding G: the number of vertices, the treewidth, and the maximum degree. More precisely, we show that any permutation group G βŠ‘ π•Š_n that can be embedded into a connected graph with m vertices, treewidth k, and maximum degree Ξ”, can also be generated by a context-free grammar of size 2^{O(kΞ”logΞ”)}β‹… m^{O(k)}. By combining our upper bound with a connection established by Pesant, Quimper, Rousseau and Sellmann [Gilles Pesant et al., 2009] between the extension complexity of a permutation group and the grammar complexity of a formal language, we also get that these permutation groups can be represented by polytopes of extension complexity 2^{O(kΞ”logΞ”)}β‹… m^{O(k)}. The above upper bounds can be used to provide trade-offs between the index of permutation groups, and the number of vertices, treewidth and maximum degree of connected graphs embedding these groups. In particular, by combining our main result with a celebrated 2^{Ξ©(n)} lower bound on the grammar complexity of the symmetric group π•Š_n due to Glaister and Shallit [Glaister and Shallit, 1996] we have that connected graphs of treewidth o(n/log n) and maximum degree o(n/log n) embedding subgroups of π•Š_n of index 2^{cn} for some small constant c must have n^{Ο‰(1)} vertices. This lower bound can be improved to exponential on graphs of treewidth n^{Ξ΅} for Ξ΅ < 1 and maximum degree o(n/log n).

BibTeX - Entry

@InProceedings{jaffke_et_al:LIPIcs:2020:12716,
  author =	{Lars Jaffke and Mateus de Oliveira Oliveira and Hans Raj Tiwary},
  title =	{{Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{50:1--50:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Javier Esparza and Daniel Kr{\'a}ΔΎ},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12716},
  URN =		{urn:nbn:de:0030-drops-127161},
  doi =		{10.4230/LIPIcs.MFCS.2020.50},
  annote =	{Keywords: Permutation Groups, Context Free Grammars, Extension Complexity, Graph Embedding Complexity}
}

Keywords: Permutation Groups, Context Free Grammars, Extension Complexity, Graph Embedding Complexity
Collection: 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
Issue Date: 2020
Date of publication: 18.08.2020


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