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DOI: 10.4230/LIPIcs.SoCG.2021.20

An Integer Programming Formulation Using Convex Polygons for the Convex Partition Problem

Authors: Hadrien Cambazard and Nicolas Catusse

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract

A convex partition of a point set P in the plane is a planar partition of the convex hull of P into empty convex polygons or internal faces whose extreme points belong to P. In a convex partition, the union of the internal faces give the convex hull of P and the interiors of the polygons are pairwise disjoint. Moreover, no polygon is allowed to contain a point of P in its interior. The problem is to find a convex partition with the minimum number of internal faces. The problem has been shown to be NP-hard and was recently used in the CG:SHOP Challenge 2020. We propose a new integer linear programming (IP) formulation that considerably improves over the existing one. It relies on the representation of faces as opposed to segments and points. A number of geometric properties are used to strengthen it. Data sets of 100 points are easily solved to optimality and the lower bounds provided by the model can be computed up to 300 points.

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Hadrien Cambazard and Nicolas Catusse. An Integer Programming Formulation Using Convex Polygons for the Convex Partition Problem. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 20:1-20:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cambazard_et_al:LIPIcs.SoCG.2021.20,
  author =	{Cambazard, Hadrien and Catusse, Nicolas},
  title =	{{An Integer Programming Formulation Using Convex Polygons for the Convex Partition Problem}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{20:1--20:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.20},
  URN =		{urn:nbn:de:0030-drops-138198},
  doi =		{10.4230/LIPIcs.SoCG.2021.20},
  annote =	{Keywords: convex partition, integer programming, geometric optimization}
}

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An Integer Programming Formulation Using Convex Polygons for the Convex Partition Problem

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