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原文摘要

In the binary diameter problem, the input consists of explicitly given permutations generating a finite group $G$ and a binary-encoded number $k$, and the question is whether every element of $G$ is a product of at most $k$ of the given generators. In the binary length problem, the input additionally contains a permutation $g \in G$, and the question is whether $g$ is such a product. We show that the binary diameter problem is PSPACE-complete, thereby confirming a conjecture of Lohrey and Rosowski (2023). Moreover, we show that PSPACE-hardness already holds for solvable groups of derived length at most 3, whereas the problem is $Π_2^{\mathrm{P}}$-complete for 2-step nilpotent groups. For the binary length problem, we establish a sharp complexity threshold: the problem is NP-complete for 2-step nilpotent groups, which extends a result of Lohrey and Rosowski for abelian groups, and PSPACE-complete for groups that are simultaneously 3-step nilpotent and metabelian, which strengthens the main result of Jerrum (1985). Finally, we show that the unary diameter problem, where $k$ is encoded in unary, is $Π_2^{\mathrm{P}}$-complete for generating sets of the full symmetric group, confirming another conjecture of Lohrey and Rosowski.