(Re)packing Equal Disks into Rectangle

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

Research output: Contribution to journalArticlepeer-review


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 languageAmerican English
JournalDiscrete and Computational Geometry
StateAccepted/In press - 1 Jan 2024


  • 51E23: Spreads and packing problems
  • 68Q25: Analysis of algorithms and problem complexity
  • 68W40: Analysis of algorithms
  • Circle packing
  • Computational geometry
  • Parameterized algorithms
  • Unit disks

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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