可解置换群的近优扩张生成集
Near-Optimal Expanding Generating Sets for Solvable Permutation Groups
摘要：
令$G =<S>$是由生成集$S$作为输入给出的对称群$S_n$的可解置换群。我们给出了一个确定性的多项式时间算法，计算一个\emph{expanding generating set}的大小为$\tilde{O}(n^2)$的$G$。更准确地说，该算法计算大小为$\tilde{O}(n^2)(1/λ)^{O(1)}$的子集$T\subset G$，使得无向凯莱图$Cay(G,T)$是$λ$ -谱扩展器（$\tilde{O}$表示法抑制$\log ^{O(1)}n$因子）。作为我们证明的副产品，我们得到了一个新的明确的结构$\varepsilon$ -偏差空间的大小为$\tilde{O}(n\poly(\log d))(\frac{1}{\varepsilon})^{O(1)}$的群体$\Z_d^n$。早期已知的大小界限是由\cite{AMN98}给出的$O((d+n/\varepsilon^2))^{11/2}$。
英文摘要：
Let $G =<S>$ be a solvable permutation group of the symmetric group $S_n$ given as input by the generating set $S$. We give a deterministic polynomial-time algorithm that computes an \emph{expanding generating set} of size $\tilde{O}(n^2)$ for $G$. More precisely, the algorithm computes a subset $T\subset G$ of size $\tilde{O}(n^2)(1/λ)^{O(1)}$ such that the undirected Cayley graph $Cay(G,T)$ is a $λ$-spectral expander (the $\tilde{O}$ notation suppresses $\log ^{O(1)}n$ factors). As a byproduct of our proof, we get a new explicit construction of $\varepsilon$-bias spaces of size $\tilde{O}(n\poly(\log d))(\frac{1}{\varepsilon})^{O(1)}$ for the groups $\Z_d^n$. The earlier known size bound was $O((d+n/\varepsilon^2))^{11/2}$ given by \cite{AMN98}.
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