Knapsack in 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 hyperbolic group, the knapsack problem can be solved in polynomial time. In this paper, it is shown that 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 effectively semilinear.