Near-Optimal Expanding Generating Sets for Solvable Permutation Groups

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Let $G =$ 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 $\tilde{O}(n^2)(1/\lambda)^{O(1)}$ such that the undirected Cayley graph $Cay(G,T)$ is a $\lambda$-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}.

💡 Research Summary

The paper addresses the algorithmic problem of constructing small expanding generating sets for solvable permutation groups. Given a solvable subgroup G ≤ Sₙ presented by a generating set S, the authors present a deterministic polynomial‑time algorithm that outputs a subset T ⊂ G of size \tilde{O}(n²)·(1/λ)^{O(1)} such that the undirected Cayley graph Cay(G,T) is a λ‑spectral expander. The notation \tilde{O} hides polylogarithmic factors in n. This result is “near‑optimal” because any generating set for a transitive subgroup of Sₙ must have size at least Ω(n), and the algorithm achieves only a polylogarithmic overhead.

The core technical contribution is a structural reduction that exploits the derived series of a solvable group. Starting from G, the algorithm computes a normal series \

Near-Optimal Expanding Generating Sets for Solvable Permutation Groups

💡 Research Summary

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