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DOI: 10.4230/LIPIcs.MFCS.2025.33

Hitting and Covering Affine Families of Convex Polyhedra, with Applications to Robust Optimization

Authors: Jean Cardinal, Xavier Goaoc, and Sarah Wajsbrot

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract

Geometric hitting set problems, in which we seek a smallest set of points that collectively hit a given set of ranges, are ubiquitous in computational geometry. Most often, the set is discrete and is given explicitly. We propose new variants of these problems, dealing with continuous families of convex polyhedra, and show that they capture decision versions of the two-level finite adaptability problem in robust optimization. We show that these problems can be solved in strongly polynomial time when the size of the hitting/covering set and the dimension of the polyhedra and the parameter space are constant. We also show that the hitting set problem can be solved in strongly quadratic time for one-parameter families of convex polyhedra in constant dimension. This leads to new tractability results for finite adaptability that are the first ones with so-called left-hand-side uncertainty, where the underlying problem is non-linear.

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Jean Cardinal, Xavier Goaoc, and Sarah Wajsbrot. Hitting and Covering Affine Families of Convex Polyhedra, with Applications to Robust Optimization. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 33:1-33:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cardinal_et_al:LIPIcs.MFCS.2025.33,
  author =	{Cardinal, Jean and Goaoc, Xavier and Wajsbrot, Sarah},
  title =	{{Hitting and Covering Affine Families of Convex Polyhedra, with Applications to Robust Optimization}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{33:1--33:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.33},
  URN =		{urn:nbn:de:0030-drops-241401},
  doi =		{10.4230/LIPIcs.MFCS.2025.33},
  annote =	{Keywords: Geometric hitting set problem, Continuous families of polyhedra, Robust optimization}
}

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Hitting and Covering Affine Families of Convex Polyhedra, with Applications to Robust Optimization

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