将置换群压缩为文法和多面体。一种图嵌入方法
Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach
摘要：
可以表明，每个置换群$G \sqsubseteq S_n$可以嵌入，在一个定义良好的意义上，在一个连接的图与$O(n+|G|)$顶点。但是，某些组需要的顶点要少得多。例如，$S_n$本身可以嵌入到$n$ -团$K_n$中，$K_n$是一个有n个顶点的连通图。在这项工作中，我们表明，生成有限置换群$G \sqsubseteq S_n$的上下文无关文法的最小尺寸可以由嵌入$G$的连通图的三个结构参数上界：顶点数，树宽，和最大程度。更准确地说，我们表明，任何置换群$G \sqsubseteq S_n$，可以嵌入到一个连接的图与$m$顶点，树宽k，和最大程度$Δ$，也可以产生的上下文无关的语法大小$2^{O(kΔ\logΔ)}\cdot m^{O(k)}$。结合置换群的扩张复杂度与形式语言的文法复杂度之间的联系，得到了置换群可以用扩张复杂度为$2^{O(k Δ\log Δ)}\cdot m^{O(k)}$的多面体表示.上面的上界可以用来提供置换群的索引与嵌入这些群的连通图的顶点数、树宽和最大度之间的折衷。特别是，通过结合我们的主要结果与一个著名的$2^{Ω(n)}$下界的语法复杂性的对称群$S_n$，我们有树宽$o(n/\log n)$和最大程度$o(n/\log n)$的连通图嵌入子群的$S_n$的指数$2^{cn}$的一些小常数$c$必须有$n^{ω(1)}$顶点。该下界可以在树宽$n^{\varepsilon}$的图上被改进为指数，用于$\varepsilon<1$和最大度$o(n/\log n)$。
英文摘要：
It can be shown that each permutation group $G \sqsubseteq S_n$ 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 $G \sqsubseteq S_n$ 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 $G \sqsubseteq S_n$ 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 $2^{O(kΔ\logΔ)}\cdot m^{O(k)}$. 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 $2^{O(k Δ\log Δ)}\cdot m^{O(k)}$. 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 $n^{\varepsilon}$ for $\varepsilon<1$ and maximum degree $o(n/\log n)$.
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