Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach
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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 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(k ΔlogΔ)· 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 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).
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