Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Lohrey, Markus, Thumm, Alexander
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214117224