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 in computational chemistry. Due to the exponentially large search space, the problem has been referred in several materials-science papers as ''NP-Hard and very challenging'' without any formal proof though. 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 from a given initial structure. 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.

中文翻译：

---

用于晶体结构预测的能量最小化硬度

晶体结构预测 (csp) 是材料科学和计算化学中的核心和最具挑战性的问题之一。在 csp 中，目标是在 3D 空间中找到一种产生最低势能的离子配置。寻找解决这个复杂优化问题的有效程序是计算化学中众所周知的开放问题。由于搜索空间呈指数级增长，该问题在几篇材料科学论文中被称为“NP-Hard 且非常具有挑战性”，但没有任何正式证据。本文填补了文献中的空白，为具有各种现实约束的 csp 变体提供了第一组正式证明的 NP 硬度结果。我们特别关注去除问题：目标是找到具有最小势能的子结构，通过从给定的初始结构中去除离子的子集。我们的主要贡献是 csp 去除问题的 NP-Hardness 结果、组合图问题到几何设置的新嵌入，以及对能量函数的更系统的探索，以揭示 csp 的复杂性。在更广泛的背景下，我们的结果有助于分析嵌入三维欧几里得空间的加权图的计算问题。