The Complexity of Knapsack in Graph Groups

TL;DR

This paper classifies the computational complexity of the knapsack problem in graph groups, showing it varies from c^0 to NP-complete depending on the graph structure.

Contribution

It precisely determines the complexity classes of the knapsack problem for all graph groups based on their underlying graph structure.

Findings

c^0-complete for complete graphs

LogCFL-complete for transitive forests

NP-complete for other graphs

Abstract

Myasnikov et al. have introduced the knapsack problem for arbitrary finitely generated groups. In previous work, the authors proved that for each graph group, the knapsack problem can be solved in $\mathsf{NP}$. Here, we determine the exact complexity of the problem for every graph group. While the problem is $\mathsf{TC}^0$-complete for complete graphs, it is $\mathsf{LogCFL}$-complete for each (non-complete) transitive forest. For every remaining graph, the problem is $\mathsf{NP}$-complete.