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dc.contributor.authorJaffke, Lars
dc.contributor.authorDe Oliveira Oliveira, Mateus
dc.contributor.authorTiwary, Hans Raj
dc.PublishedLeibniz International Proceedings in Informatics. 2020, 170 1-15.
dc.description.abstractIt 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.publisherDagstuhl publishingen_US
dc.rightsNavngivelse 4.0 Internasjonal*
dc.titleCompressing permutation groups into grammars and polytopes. A graph embedding approachen_US
dc.typeJournal articleen_US
dc.typePeer revieweden_US
dc.rights.holderCopyright 2020 The Authorsen_US
dc.source.journalLeibniz International Proceedings in Informaticsen_US
dc.identifier.citationLeibniz International Proceedings in Informatics. 2020, 50, 50:1–50:15en_US

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