Knapsack and subset sum problems in nilpotent, polycyclic, and co-context-free groups

TL;DR

This paper investigates the decidability and complexity of the knapsack and subset sum problems across various classes of groups, revealing undecidability in some cases and efficient solutions in others, with implications for group theory and computational complexity.

Contribution

It demonstrates undecidability of the knapsack problem in certain nilpotent groups and decidability in others, and establishes complexity bounds for subset sum in polycyclic and virtually nilpotent groups.

Findings

Knapsack problem is undecidable in direct products of many Heisenberg groups.

Knapsack problem is decidable in the Heisenberg group itself.

Subset sum problem is in NL for finitely generated virtually nilpotent groups.

Abstract

It is shown that the knapsack problem (introduced by Myasnikov, Nikolaev, and Ushakov) is undecidable in a direct product of sufficiently many copies of the discrete Heisenberg group (which is nilpotent of class 2). Moreover, for the discrete Heisenberg group itself, the knapsack problem is decidable. Hence, decidability of the knapsack problem is not preserved under direct products. It is also shown that for every co-context-free group, the knapsack problem is decidable. For the subset sum problem (also introduced by Myasnikov, Nikolaev, and Ushakov) we show that it belongs to the class NL (nondeterministic logspace) for every finitely generated virtually nilpotent group and that there exists a polycyclic group with an NP-complete subset sum problem.