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: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12716/
Jaffke, Lars ;
de Oliveira Oliveira, Mateus ;
Tiwary, Hans Raj
Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach
Abstract
It can be shown that each permutation group G ⊑ ?_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, ?_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 G⊑ ?_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 ⊑ ?_n that can be embedded into a connected graph with m vertices, treewidth k, and maximum degree Δ, can also be generated by a context-free grammar of size 2^{O(kΔlogΔ)}⋅ 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(kΔlogΔ)}⋅ 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^{Ω(n)} lower bound on the grammar complexity of the symmetric group ?_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 ?_n of index 2^{cn} for some small constant c must have n^{ω(1)} vertices. This lower bound can be improved to exponential on graphs of treewidth n^{ε} for ε < 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}ľ},
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}
}
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Keywords: |
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Permutation Groups, Context Free Grammars, Extension Complexity, Graph Embedding Complexity |
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Collection: |
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45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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18.08.2020 |
