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Approximating the Maximum Independent Set of Convex Polygons with a Bounded Number of Directions

Authors Fabrizio Grandoni , Edin Husić , Mathieu Mari , Antoine Tinguely

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ACM Subject Classification
  • Theory of computation → Packing and covering problems
  • Theory of computation → Computational geometry
Keywords
  • Approximation algorithms
  • packing
  • independent set
  • polygons

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Abstract

In the maximum independent set of convex polygons problem, we are given a set of n convex polygons in the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons. Here we study a special case of the problem where the edges of the polygons can take at most d fixed directions. We present an 8d/3-approximation algorithm for this problem running in time O((nd)^O(d4^d)). The previous-best polynomial-time approximation (for constant d) was a classical n^ε approximation by Fox and Pach [SODA'11] that has recently been improved to a OPT^ε-approximation algorithm by Cslovjecsek, Pilipczuk and Węgrzycki [SODA '24], which also extends to an arbitrary set of convex polygons.
Our result builds on, and generalizes the recent constant factor approximation algorithms for the maximum independent set of axis-parallel rectangles problem (which is a special case of our problem with d = 2) by Mitchell [FOCS'21] and Gálvez, Khan, Mari, Mömke, Reddy, and Wiese [SODA'22].

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Fabrizio Grandoni, Edin Husić, Mathieu Mari, and Antoine Tinguely. Approximating the Maximum Independent Set of Convex Polygons with a Bounded Number of Directions. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 61:1-61:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.61

Author Details

Fabrizio Grandoni
  • IDSIA, USI-SUPSI, Lugano, Switzerland
Edin Husić
  • IDSIA, USI-SUPSI, Lugano, Switzerland
Mathieu Mari
  • LIRMM, University of Montpellier, CNRS, Montpellier, France
Antoine Tinguely
  • IDSIA, USI-SUPSI, Lugano, Switzerland

Funding

  • Grandoni, Fabrizio: Partially supported by the Swiss National Science Foundation (SNSF) Grant 200021 200731/1.
  • Husić, Edin: Supported by the Swiss National Science Foundation (SNSF) Grant 200021 200731/1.
  • Tinguely, Antoine: Supported by the Swiss National Science Foundation (SNSF) Grant 200021 200731/1.

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