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Covering and Partitioning Complex Objects with Small Pieces

Authors Anders Aamand , Mikkel Abrahamsen , Reilly Browne , Mayank Goswami , Prahlad Narasimhan Kasthurirangan , Linda Kleist , Joseph S. B. Mitchell , Valentin Polishchuk , Jack Stade

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Packing and covering problems
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Covering
  • partitioning
  • polygon
  • small piece
  • PTAS

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Abstract

We study the problems of covering or partitioning a polygon P (possibly with holes) using a minimum number of small pieces, where a small piece is a connected sub-polygon contained in an axis-aligned unit square. For covering, we seek to write P as a union of small pieces, and in partitioning, we furthermore require the pieces to be pairwise interior-disjoint. We show that these problems are in fact equivalent: Optimum covers and partitions have the same number of pieces.
For covering, a natural local search algorithm repeatedly attempts to replace k pieces from a candidate cover with k-1 pieces. In two dimensions and for sufficiently large k, we show that when no such swap is possible, the cover is a 1+ O(1/√k) approximation, hence obtaining the first PTAS for the problem. Prior to our work, the only known algorithm was a 13-approximation that only works for polygons without holes [Abrahamsen and Rasmussen, SODA 2025]. In contrast, in the three dimensional version of the problem, for a polyhedron P of complexity n, we show that it is NP-hard to approximate an optimal cover or partition to within a factor that is logarithmic in n, even if P is simple, i.e., has genus 0 and no holes.

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Anders Aamand, Mikkel Abrahamsen, Reilly Browne, Mayank Goswami, Prahlad Narasimhan Kasthurirangan, Linda Kleist, Joseph S. B. Mitchell, Valentin Polishchuk, and Jack Stade. Covering and Partitioning Complex Objects with Small Pieces. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.1

Author Details

Anders Aamand
  • Department of Computer Science, University of Copenhagen, Denmark
Mikkel Abrahamsen
  • Department of Computer Science, University of Copenhagen, Denmark
Reilly Browne
  • Department of Computer Science, Dartmouth College, Hanover, NH, USA
Mayank Goswami
  • Department of Computer Science, Queens College CUNY, New York, NY, USA
Prahlad Narasimhan Kasthurirangan
  • Department of Applied Mathematics and Statistics, Stony Brook University, NY, USA
Linda Kleist
  • Department of Informatics, Universtity Hamburg, Germany
Joseph S. B. Mitchell
  • Department of Applied Mathematics and Statistics, Stony Brook University, NY, USA
Valentin Polishchuk
  • Communications and Transport Systems, Linköping Univeristy, Sweden
Jack Stade
  • Department of Computer Science, University of Copenhagen, Denmark

Funding

  • Aamand, Anders: This work was supported by the VILLUM Foundation grant 54451.
  • Abrahamsen, Mikkel: Supported by Independent Research Fund Denmark, grant 1054-00032B, and by the Carlsberg Foundation, grant CF24-1929.
  • Goswami, Mayank: Supported by National Science Foundation CCF-2503086.
  • Kasthurirangan, Prahlad Narasimhan: Some of this work was carried out during a research stay at TU Braunschweig. Partially supported by the US Office of Naval Research under grant no. N00014-26-1-2088.
  • Stade, Jack: Supported by Independent Research Fund Denmark, grant 1054-00032B, and by the Carlsberg Foundation, grant CF24-1929.

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