Knapsack in hyperbolic groups

TL;DR

This paper precisely characterizes the computational complexity of the knapsack problem in hyperbolic groups, showing it is in LogCFL and providing semilinear representations of solutions, extending to certain group classes.

Contribution

It determines the exact complexity class of the knapsack problem for hyperbolic groups and introduces the concept of knapsack-tame groups with polynomial-size semilinear solutions.

Findings

Knapsack problem in hyperbolic groups is LogCFL-complete if the group contains a free subgroup of rank two.

The set of solutions forms a semilinear set with polynomial-size representations.

Knapsack problem remains in LogCFL for groups formed by free and direct products with hyperbolic groups.

Abstract

Recently knapsack problems have been generalized from the integers to arbitrary finitely generated groups. The knapsack problem for a finitely generated group $G$ is the following decision problem: given a tuple $(g, g_1, \ldots, g_k)$ of elements of $G$, are there natural numbers $n_1, \ldots, n_k \in \mathbb{N}$ such that $g = g_1^{n_1} \cdots g_k^{n_k}$ holds in $G$? Myasnikov, Nikolaev, and Ushakov proved that for every (Gromov-)hyperbolic group, the knapsack problem can be solved in polynomial time. In this paper, the precise complexity of the knapsack problem for hyperbolic group is determined: for every hyperbolic group $G$, the knapsack problem belongs to the complexity class $\mathsf{LogCFL}$, and it is $\mathsf{LogCFL}$-complete if $G$ contains a free group of rank two. Moreover, it is shown that for every hyperbolic group $G$ and every tuple $(g, g_1, \ldots, g_k)$ of elements of $G$ the set of all $(n_1, \ldots, n_k) \in \mathbb{N}^k$ such that $g = g_1^{n_1} \cdots g_k^{n_k}$ in $G$ is semilinear and a semilinear representation where all integers are of size polynomial in the total geodesic length of the $g, g_1, \ldots, g_k$ can be computed. Groups with this property are also called knapsack-tame. This enables us to show that knapsack can be solved in $\mathsf{LogCFL}$ for every group that belongs to the closure of hyperbolic groups under free products and direct products with $\mathbb{Z}$.