(Re)packing Equal Disks into Rectangle

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Meirav Zehavi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We consider the following algorithmic generalization of the equal disk packing problem. In this problem, for a given packing of equal disks into a rectangle, the question is whether by changing positions of a small number of disks, we can allocate space for packing more disks. More formally, in the repacking problem, for a given set of n equal disks packed into a rectangle and integers k and h, we ask whether it is possible by changing positions of at most h disks to pack n + k disks. Thus the problem of packing equal disks is the special case of our problem with n = h = 0. While the computational complexity of packing equal disks into a rectangle remains open, we prove that the repacking problem is NP-hard already for h = 0. Our main algorithmic contribution is an algorithm that solves the repacking problem in time (h + k)O(h+k) · |I|O(1), where |I| is the input size. That is, the problem is fixed-parameter tractable parameterized by k and h.

Original languageEnglish
Title of host publication49th EATCS International Conference on Automata, Languages, and Programming, ICALP 2022
EditorsMikolaj Bojanczyk, Emanuela Merelli, David P. Woodruff
Place of PublicationDagstuhl, German
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages60:1-60:17
Number of pages17
Volume229
ISBN (Electronic)9783959772358
ISBN (Print)978-3-95977-235-8
DOIs
StatePublished - 2022
Event49th EATCS International Conference on Automata, Languages, and Programming, ICALP 2022 - Paris, France
Duration: 4 Jul 20228 Jul 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume229
ISSN (Print)1868-8969

Conference

Conference49th EATCS International Conference on Automata, Languages, and Programming, ICALP 2022
Country/TerritoryFrance
CityParis
Period4/07/228/07/22

Keywords

  • circle packing
  • fixed-parameter tractability
  • parameterized complexity
  • unit disks

ASJC Scopus subject areas

  • Software

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