The complexity of knapsack problems in wreath products

Abstract

We prove new complexity results for computational problems in certain wreath products of groups and (as an application) for free solvable group. For a finitely generated group we study the so-called power word problem (does a given expression u k 1 1 . . . u k d d , where u1, . . . , u d are words over the group generators and k1, . . . , k d are binary encoded integers, evaluate to the group identity?) and knapsack problem (does a given equation u x 1 1 . . . u x d d = v, where u1, . . . , u d , v are words over the group generators and x1, . . . , x d are variables, has a solution in the natural numbers). We prove that the power word problem for wreath products of the form G Z with G nilpotent and iterated wreath products of free abelian groups belongs to TC 0 . As an application of the latter, the power word problem for free solvable groups is in TC 0 . On the other hand we show that for wreath products G Z, where G is a so called uniformly strongly efficiently non-solvable group (which form a large subclass of non-solvable groups), the power word problem is coNP-hard. For the knapsack problem we show NP-completeness for iterated wreath products of free abelian groups and hence free solvable groups. Moreover, the knapsack problem for every wreath product G Z, where G is uniformly efficiently non-solvable, is Σ 2 p -hard.2012 ACM Subject Classification CCS → Theory of computation → computational complexity and cryptography → problems, reductions and completeness Keywords and phrases algorithmic group theory, knapsack, wreath product