Knapsack Problems in Groups

TL;DR

This paper extends classical knapsack and subset sum problems to arbitrary groups, analyzing their computational complexity and identifying cases where they are efficiently decidable or NP-complete.

Contribution

It introduces group-based versions of knapsack problems and characterizes their complexity across different classes of groups, including hyperbolic and finitely presented groups.

Findings

Knapsack problems are P-time decidable in hyperbolic groups.

Subset sum problem is NP-complete in some finitely presented groups.

Bounded submonoid membership problem shares similar complexity results.

Abstract

We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems. We show that these problems, as well as the bounded submonoid membership problem, are P-time decidable in hyperbolic groups and give various examples of finitely presented groups where the subset sum problem is NP-complete.