License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2022.8
URN: urn:nbn:de:0030-drops-168065
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16806/
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Adamson, Duncan ; Deligkas, Argyrios ; Gusev, Vladimir V. ; Potapov, Igor

The Complexity of Periodic Energy Minimisation

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Abstract

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.

BibTeX - Entry

@InProceedings{adamson_et_al:LIPIcs.MFCS.2022.8,
  author =	{Adamson, Duncan and Deligkas, Argyrios and Gusev, Vladimir V. and Potapov, Igor},
  title =	{{The Complexity of Periodic Energy Minimisation}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{8:1--8:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16806},
  URN =		{urn:nbn:de:0030-drops-168065},
  doi =		{10.4230/LIPIcs.MFCS.2022.8},
  annote =	{Keywords: Optimisation of periodic structures, tiling, undecidability, NP-hardness}
}

Keywords: Optimisation of periodic structures, tiling, undecidability, NP-hardness
Collection: 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)
Issue Date: 2022
Date of publication: 22.08.2022

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