Word Length and Diameter in Permutation Groups

2026-09-22

AI Summary

研究解决了置换群中的二进制直径和长度问题，通过证明其在不同条件下的计算复杂性。

Abstract

The input for the binary diameter problem consists of explicitly represented permutations generating a finite group $G$ and a binary-encoded nonnegative integer $k$. The question is whether every element of $G$ is a product of at most $k$ input generators. For the binary length problem, the input contains in addition a permutation $g \in G$ and it is asked whether $g$ is a product of at most $k$ input generators. We prove that the binary diameter problem is PSPACE-complete. When restricted to $2$-step nilpotent groups, the binary diameter problem is shown to be complete for $\mathsf{Π_2^P}$, whereas the binary length problem is shown to be NP-complete. Without the restriction to $2$-step nilpotent groups, the binary length problem is PSPACE-complete by a result of Jerrum.

Problem

• Permutation Groups
• Binary Diameter Problem
• Binary Length Problem
• PSPACE-complete
• NP-complete

Innovation

• PSPACE-complete
• Π₂^P-complete
• NP-complete
• binary diameter problem
• binary length problem