(Re)packing Equal Disks into Rectangle

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 $n$ 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 $(h+k)^{O(h+k)}\cdot |I|^{O(1)}$, where $I$ is the input size. That is, the problem is fixed-parameter tractable parameterized by $k$ and $h$.