(Re)packing Equal Disks into Rectangle
Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh,, Meirav Zehavi

TL;DR
This paper introduces a generalized disk packing problem, proves its NP-hardness, and presents a fixed-parameter tractable algorithm for solving it efficiently based on parameters k and h.
Contribution
It formalizes the repacking problem for equal disks, proves NP-hardness for the case h=0, and develops a fixed-parameter tractable algorithm.
Findings
Repacking problem is NP-hard for h=0.
Proposed algorithm runs in time (h+k)^{O(h+k)}·|I|^{O(1)}.
Problem is fixed-parameter tractable with respect to k and h.
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 equal disks packed into a rectangle and integers and , we ask whether it is possible by changing positions of at most disks to pack disks. Thus the problem of packing equal disks is the special case of our problem with . While the computational complexity of packing equal disks into a rectangle remains open, we prove that the repacking…
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Taxonomy
TopicsOptimization and Packing Problems · Computational Geometry and Mesh Generation · Product Development and Customization
