dc.contributor.author Jaffke, Lars dc.contributor.author De Oliveira Oliveira, Mateus dc.contributor.author Tiwary, Hans Raj dc.date.accessioned 2021-06-24T09:27:17Z dc.date.available 2021-06-24T09:27:17Z dc.date.created 2020-09-22T16:27:57Z dc.date.issued 2020 dc.Published Leibniz International Proceedings in Informatics. 2020, 170 1-15. dc.identifier.issn 1868-8969 dc.identifier.uri https://hdl.handle.net/11250/2761072 dc.description.abstract It can be shown that each permutation group G ⊑ 𝕊_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, 𝕊_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⊑ 𝕊_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 ⊑ 𝕊_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 𝕊_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 𝕊_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). en_US dc.language.iso eng en_US dc.publisher Dagstuhl publishing en_US dc.rights Navngivelse 4.0 Internasjonal * dc.rights.uri http://creativecommons.org/licenses/by/4.0/deed.no * dc.title Compressing permutation groups into grammars and polytopes. A graph embedding approach en_US dc.type Journal article en_US dc.type Peer reviewed en_US dc.description.version publishedVersion en_US dc.rights.holder Copyright 2020 The Authors en_US cristin.ispublished true cristin.fulltext original cristin.qualitycode 1 dc.identifier.doi 10.4230/LIPIcs.MFCS.2020.50 dc.identifier.cristin 1832258 dc.source.journal Leibniz International Proceedings in Informatics en_US dc.source.40 170 dc.source.pagenumber 1-15 en_US dc.identifier.citation Leibniz International Proceedings in Informatics. 2020, 50, 50:1–50:15 en_US dc.source.volume 50 en_US
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