Affiliations: [a] Department of Computer Science, University of Liverpool, Liverpool, UK. duncan.adamson@liverpool.ac.uk | [b] Department of Computer Science, Royal Holloway University of London, London, UK. Argyrios.Deligkas@rhul.ac.uk | [c] Leverhulme Research Centre for Functional Materials Design, University of Liverpool, Liverpool, UK. Vladimir.Gusev@liverpool.ac.uk | [d] Department of Computer Science, University of Liverpool, Liverpool, UK. potapov@liverpool.ac.uk
Correspondence:
[†]
Address for correspondence: Department of Computer Science, University of Liverpool, Liverpool, L69 3BX, UK.
Note: [*] A preliminary conference version of this work appeared in Proceedings of the 46th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2020 [1].
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.
