The Complexity of Periodic Energy Minimisation

Authors Duncan Adamson, Argyrios Deligkas, Vladimir V. Gusev, Igor Potapov

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Duncan Adamson
  • ICE-TCS, Department of Computer Science, Reykjavik University, Iceland
Argyrios Deligkas
  • Department of Computer Science, Royal Holloway, University of London, UK
Vladimir V. Gusev
  • Materials Innovation Factory, Department of Computer Science, University of Liverpool, UK
Igor Potapov
  • Department of Computer Science, University of Liverpool, UK

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Duncan Adamson, Argyrios Deligkas, Vladimir V. Gusev, and Igor Potapov. The Complexity of Periodic Energy Minimisation. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


The computational complexity of pairwise energy minimisation of N points in real space is a long-standing open problem. The idea of the potential intractability of the problem was supported by a lack of progress in finding efficient algorithms, even when restricted the integer grid approximation. In this paper we provide a firm answer to the problem on ℤ^d by showing that for a large class of pairwise energy functions the problem of periodic energy minimisation is NP-hard if the size of the period (known as a unit cell) is fixed, and is undecidable otherwise. We do so by introducing an abstraction of pairwise average energy minimisation as a mathematical problem, which covers many existing models. The most influential aspects of this work are showing for the first time: 1) undecidability of average pairwise energy minimisation in general 2) computational hardness for the most natural model with periodic boundary conditions, and 3) novel reductions for a large class of generic pairwise energy functions covering many physical abstractions at once. In particular, we develop a new tool of overlapping digital rhombuses to incorporate the properties of the physical force fields, and we connect it with classical tiling problems. Moreover, we illustrate the power of such reductions by incorporating more physical properties such as charge neutrality, and we show an inapproximability result for the extreme case of the 1D average energy minimisation problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Optimisation of periodic structures
  • tiling
  • undecidability
  • NP-hardness


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