@inproceedings{e4d3a83a0357494da86ab36807125ded,

title = "(Re)packing Equal Disks into Rectangle",

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.",

keywords = "circle packing, fixed-parameter tractability, parameterized complexity, unit disks",

author = "Fomin, {Fedor V.} and Golovach, {Petr A.} and Tanmay Inamdar and Meirav Zehavi",

note = "Funding Information: Funding Fedor V. Fomin: Supported by the Research Council of Norway via the project BWCA, grant no. 314528. Petr A. Golovach: Supported by the Research Council of Norway via the project BWCA, grant no. 314528. Tanmay Inamdar: Supported by the European Research Council (ERC) via grant LOPPRE, reference 819416. Meirav Zehavi: Supported by the Israel Science Foundation (ISF) grant no. 1176/18. Publisher Copyright: {\textcopyright} Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, and Meirav Zehavi; licensed under Creative Commons License CC-BY 4.0; 49th EATCS International Conference on Automata, Languages, and Programming, ICALP 2022 ; Conference date: 04-07-2022 Through 08-07-2022",

year = "2022",

month = jun,

day = "28",

doi = "https://doi.org/10.4230/LIPIcs.ICALP.2022.60",

language = "American English",

isbn = "978-3-95977-235-8",

volume = "229",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

pages = "60:1--60:17",

editor = "Mikolaj Bojanczyk and Emanuela Merelli and Woodruff, {David P.}",

booktitle = "49th EATCS International Conference on Automata, Languages, and Programming, ICALP 2022",

address = "Germany",

}