Complexity of Spherical Equations in Finite Groups

TL;DR

This paper explores the computational complexity of solving spherical equations in finite groups, revealing polynomial-time solvability for some groups and NP-completeness for others, depending on the group class and input parameters.

Contribution

It classifies the complexity of the Diophantine problem for spherical equations across various finite groups, identifying cases of polynomial-time solvability and NP-completeness.

Findings

Polynomial-time solvability for fixed or table-encoded groups.

NP-completeness for permutation groups $S_n$ with $n$ as input.

Mixed complexity results for matrix groups, including NP-complete cases and polynomial solutions for $GL(2,p)$.

Abstract

In this paper we investigate computational properties of the Diophantine problem for spherical equations in some classes of finite groups. We classify the complexity of different variations of the problem, e.g., when $G$ is fixed and when $G$ is a part of the input. When the group $G$ is constant or given as multiplication table, we show that the problem always can be solved in polynomial time. On the other hand, for the permutation groups $S_n$ (with $n$ part of the input), the problem is NP-complete. The situation for matrix groups is quite involved: while we exhibit sequences of 2-by-2 matrices where the problem is NP-complete, in the full group $GL(2,p)$ ($p$ prime and part of the input) it can be solved in polynomial time. We also find a similar behaviour with subgroups of matrices of arbitrary dimension over a constant ring.