Subset sum problem in polycyclic groups

TL;DR

This paper investigates the computational complexity of the subset sum problem within polycyclic groups, revealing a transition from polynomial-time solvability in virtually nilpotent groups to NP-completeness in non-virtually-nilpotent cases.

Contribution

It demonstrates that subgroup distortion can be used to establish NP-completeness of subset sum in non-virtually-nilpotent polycyclic groups, extending complexity results in group theory.

Findings

Subset sum is polynomial-time decidable in virtually nilpotent groups.

Subset sum becomes NP-complete in non-virtually-nilpotent polycyclic groups.

Subgroup distortion is a key tool in proving complexity results.

Abstract

We consider a group-theoretic analogue of the classic subset sum problem. It is known that every virtually nilpotent group has polynomial time decidable subset sum problem. In this paper we use subgroup distortion to show that every polycyclic non-virtually-nilpotent group has NP-complete subset sum problem.