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,g1,…,gk) of elements of G, are there natural numbers n1,…,nk∈N such that g=g1n1⋯gknk 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 LogCFL, and it is 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,g1,…,gk) of elements of G the set of all (n1,…,nk)∈Nk such that g=g1n1⋯gknk in G is semilinear and a semilinear representation where all integers are of size polynomial in the total geodesic length of the g,g1,…,gk can be computed. Groups with this property are also called knapsack-tame. This enables us to show that knapsack can be solved in LogCFL for every group that belongs to the closure of hyperbolic groups under free products and direct products with Z.