eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-28
60:1
60:17
10.4230/LIPIcs.ICALP.2022.60
article
(Re)packing Equal Disks into Rectangle
Fomin, Fedor V.
1
https://orcid.org/0000-0003-1955-4612
Golovach, Petr A.
1
https://orcid.org/0000-0002-2619-2990
Inamdar, Tanmay
1
https://orcid.org/0000-0002-0184-5932
Zehavi, Meirav
2
https://orcid.org/0000-0002-3636-5322
Department of Informatics, University of Bergen, Norway
Ben-Gurion University of the Negev, Beer-Sheva, Israel
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol229-icalp2022/LIPIcs.ICALP.2022.60/LIPIcs.ICALP.2022.60.pdf
circle packing
unit disks
parameterized complexity
fixed-parameter tractability