On the Hardness of Energy Minimisation for Crystal Structure Prediction

TL;DR

This paper proves that the problem of finding minimal energy substructures in crystal structure prediction is NP-Hard, establishing its computational difficulty with formal proofs and embedding graph problems into geometric settings.

Contribution

It provides the first formal NP-Hardness proofs for a variant of crystal structure prediction involving ion removal, advancing understanding of its computational complexity.

Findings

NP-Hardness of the csp removal problem established

New embeddings of graph problems into 3D geometric space

Systematic analysis of the energy function complexity

Abstract

Crystal Structure Prediction (csp) is one of the central and most challenging problems in materials science and computational chemistry. In csp, the goal is to find a configuration of ions in 3D space that yields the lowest potential energy. Finding an efficient procedure to solve this complex optimisation question is a well known open problem. Due to the exponentially large search space, the problem has been referred in several materials-science papers as "NP-Hard and very challenging" without a formal proof. This paper fills a gap in the literature providing the first set of formally proven NP-Hardness results for a variant of csp with various realistic constraints. In particular, we focus on the problem of removal: the goal is to find a substructure with minimal potential energy, by removing a subset of the ions. Our main contributions are NP-Hardness results for the csp removal problem, new embeddings of combinatorial graph problems into geometrical settings, and a more systematic exploration of the energy function to reveal the complexity of csp. In a wider context, our results contribute to the analysis of computational problems for weighted graphs embedded into the three-dimensional Euclidean space.