NP-hardness of the cluster minimization problem revisited

TL;DR

This paper revisits the NP-hardness of the cluster minimization problem, providing a new proof for physically relevant potentials and discussing limitations and related subproblems for cluster analysis.

Contribution

It offers a revised NP-hardness proof applicable to geometric potentials and introduces related subproblems relevant to cluster numerical studies.

Findings

Original NP-hardness proof does not apply to geometric potentials.

A new polynomial-time reduction from the independent set problem is presented.

Limitations of the geometric formulation are discussed.

Abstract

The computational complexity of the "cluster minimization problem" is revisited [L. T. Wille and J. Vennik, J. Phys. A 18, L419 (1985)]. It is argued that the original NP-hardness proof does not apply to pairwise potentials of physical interest, such as those that depend on the geometric distance between the particles. A geometric analog of the original problem is formulated, and a new proof for such potentials is provided by polynomial time transformation from the independent set problem for unit disk graphs. Limitations of this formulation are pointed out, and new subproblems that bear more direct consequences to the numerical study of clusters are suggested.