The Power Word Problem in Graph Products

TL;DR

This paper investigates the computational complexity of the power word problem in graph products of groups, establishing reductions to known problems and correcting previous results by adding the assumption that base groups lack elements of order two.

Contribution

It introduces new complexity bounds for the power word problem in graph products, especially under the assumption that base groups have no elements of order two, and corrects earlier inaccuracies.

Findings

Power word problem for a group reduces to that of a finite-index subgroup.

In fixed graph products without elements of order two, the problem reduces to the word problem for free groups.

Uniform power word problem solvable in a specific complexity class, leading to NP-completeness of the uniform knapsack problem in right-angled Artin groups.

Abstract

The power word problem for a group $G$ asks whether an expression $u_1^{x_1} \cdots u_n^{x_n}$, where the $u_i$ are words over a finite set of generators of $G$ and the $x_i$ binary encoded integers, is equal to the identity of $G$. It is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over $G$). We start by showing some easy results concerning the power word problem. In particular, the power word problem for a group $G$ is $NC^1$-many-one reducible to the power word problem for a finite-index subgroup of $G$. For our main result, we consider graph products of groups that do not have elements of order two. We show that the power word problem in a fixed such graph product is $AC^0$-Turing-reducible to the word problem for the free group $F_2$ and the power word problems of the base groups. Furthermore, we look into the uniform power word problem in a graph product, where the dependence graph and the base groups are part of the input. Given a class of finitely generated groups $\mathcal{C}$ without order two elements, the uniform power word problem in a graph product can be solved in $\mathsf{AC^0(C_=L^{UPowWP(\mathcal{C})})}$, where $UPowWP(\mathcal{C})$ denotes the uniform power word problem for groups from the class $\mathcal{C}$. As a consequence of our results, the uniform knapsack problem in right-angled Artin groups is NP-complete. The present paper is a combination of the two conference papers. In [StoberW22] and previous iterations of this paper our results on graph products were wrongly stated without the additional assumption that the base groups do not have elements of order two. In the present work we correct this mistake. While we strongly conjecture that the result as stated previously is true, our proof relies on this additional assumption.