文章基本信息

标题：Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach
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作者：Lars Jaffke ; Mateus de Oliveira Oliveira ; Hans Raj Tiwary 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：170
页码：50:1-50:15
DOI：10.4230/LIPIcs.MFCS.2020.50
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：It can be shown that each permutation group G âS' ð.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, ð.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 GâS' ð.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 âS' ð.S_n that can be embedded into a connected graph with m vertices, treewidth k, and maximum degree Î", can also be generated by a context-free grammar of size 2^{O(kÎ"logÎ")}â<. 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(kÎ"logÎ")}â<. 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^{Î©(n)} lower bound on the grammar complexity of the symmetric group ð.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 ð.S_n of index 2^{cn} for some small constant c must have n^{Ï(1)} vertices. This lower bound can be improved to exponential on graphs of treewidth n^{Îµ} for Îµ < 1 and maximum degree o(n/log n).
关键词：Permutation Groups; Context Free Grammars; Extension Complexity; Graph Embedding Complexity