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Minimum Partition of Polygons Under Width and Cut Constraints

Authors Jaehoon Chung , Kazuo Iwama, Chung-Shou Liao, Hee-Kap Ahn

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Polygon partitioning
  • Width constraints
  • Plank problem

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Abstract

We study the problem of partitioning a polygon into the minimum number of subpolygons using cuts in predetermined directions such that each resulting subpolygon satisfies a given width constraint. A polygon satisfies the unit-width constraint for a set of unit vectors if the length of the orthogonal projection of the polygon on a line parallel to a vector in the set is at most one. We analyze structural properties of the minimum partition numbers, focusing on monotonicity under polygon containment. We show that the minimum partition number of a simple polygon is at least that of any subpolygon, provided that the subpolygon satisfies a certain orientation-wise convexity with respect to the polygon. As a consequence, we prove a partition analogue of the Bang’s conjecture about coverings of convex regions in the plane: for any partition of a convex body in the plane, the sum of relative widths of all parts is at least one. For any convex polygon, there exists a direction along which an optimal partition is achieved by parallel cuts. Given such a direction, an optimal partition can be computed in linear time.

Cite As Get BibTex

Jaehoon Chung, Kazuo Iwama, Chung-Shou Liao, and Hee-Kap Ahn. Minimum Partition of Polygons Under Width and Cut Constraints. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.22

Author Details

Jaehoon Chung
  • Korea Institute for Advanced Study (KIAS), Seoul, South Korea
Kazuo Iwama
  • Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, Taiwan
Chung-Shou Liao
  • Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan
Hee-Kap Ahn
  • Graduate School of Artificial Intelligence, Department of Computer Science and Engineering, Pohang University of Science and Technology (POSTECH), South Korea

Funding

  • Chung, Jaehoon: Supported by a KIAS Individual Grant AP106101 via the Center for Artificial Intelligence and Natural Sciences at Korea Institute for Advanced Study.
  • Iwama, Kazuo: Supported in part by MOST, Taiwan, under Grants NSTC 110-2223-E-007-001 and NSTC 111-2223-E-007-010.
  • Liao, Chung-Shou: Supported by MOST Taiwan Grants NSTC 111-2221-E-002-207-MY3, 114-2221-E-002-221-MY3, and 114-2221-E-002-220-MY3.
  • Ahn, Hee-Kap: Supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (RS-2023-00219980) and the Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) (No. RS-2019-II191906, Artificial Intelligence Graduate School Program(POSTECH)).

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