For a finite group $G$, the size of a minimum generating set of $G$ is denoted by $d(G)$. Given a finite group $G$ and an integer $k$, deciding if $d(G)\leq k$ is known as the minimum generating set (MIN-GEN) problem. A group $G$ of order $n$ has generating set of size $\lceil \log_p n \rceil$ where $p$ is the smallest prime dividing $n=|G|$. This fact is used to design an $n^{\log_p n+O(1)}$-time algorithm for the group isomorphism problem of groups specified by their Cayley tables (attributed to Tarjan by Miller, 1978). The same fact can be used to give an $n^{\log_p n+O(1)}$-time algorithm for the MIN-GEN problem. We show that the MIN-GEN problem can be solved in time $n^{(1/4)\log_p n+O(1)}$ for general groups given by their Cayley tables. This runtime incidentally matches with the runtime of the best known algorithm for the group isomorphism problem. We show that if a group $G$, given by its Cayley table, is the product of simple groups then a minimum generating set of $G$ can be computed in time polynomial in $|G|$. Given groups $G_i$ along with $d(G_i)$ for $i\in [r]$ the problem of computing $d(\Pi_{i\in[r]} G_i)$ is nontrivial. As a consequence of our result for products of simple groups we show that this problem also can be solved in polynomial time for Cayley table representation. For the MIN-GEN problem for permutation groups, to the best of our knowledge, no significantly better algorithm than the brute force algorithm is known. For an input group $G\leq S_n$, the brute force algorithm runs in time $|G|^{O(n)}$ which can be $2^{\Omega(n^2)}$. We show that if $G\leq S_n$ is a primitive permutation group then the MIN-GEN problem can be solved in time quasi-polynomial in $n$. We also design a $\mathrm{DTIME}(2^n)$ algorithm for computing a minimum generating set of permutation groups all of whose non-abelian chief factors have bounded orders.

翻译：对于有限群 $G$，其最小生成集的大小记为 $d(G)$。给定有限群 $G$ 和整数 $k$，判断是否 $d(G)\leq k$ 称为最小生成集（MIN-GEN）问题。阶为 $n$ 的群 $G$ 存在大小为 $\lceil \log_p n \rceil$ 的生成集，其中 $p$ 是整除 $n=|G|$ 的最小素数。这一结论被用于设计基于凯莱表的群同构问题（归功于 Miller, 1978 中的 Tarjan）的 $n^{\log_p n+O(1)}$ 时间算法。同样该结论可用于给出 MIN-GEN 问题的 $n^{\log_p n+O(1)}$ 时间算法。我们证明，对于以凯莱表给出的一般群，MIN-GEN 问题可在 $n^{(1/4)\log_p n+O(1)}$ 时间内求解。该运行时间恰好与群同构问题已知最优算法的运行时间一致。我们证明，若以凯莱表给出的群 $G$ 是单群的直积，则可在 $|G|$ 的多项式时间内计算 $G$ 的一个最小生成集。给定群 $G_i$ 及其 $d(G_i)$（$i\in [r]$），计算 $d(\Pi_{i\in[r]} G_i)$ 的问题是非平凡的。基于我们对单群直积的结果，我们证明对于凯莱表表示，该问题也可在多项式时间内求解。对于置换群的 MIN-GEN 问题，据我们所知，尚不存在显著优于暴力搜索的算法。对于输入群 $G\leq S_n$，暴力搜索算法的运行时间为 $|G|^{O(n)}$，可能达到 $2^{\Omega(n^2)}$。我们证明，若 $G\leq S_n$ 是本原置换群，则 MIN-GEN 问题可在 $n$ 的拟多项式时间内求解。我们还设计了一个 $\mathrm{DTIME}(2^n)$ 算法，用于计算所有非交换主因子具有有界阶的置换群的最小生成集。