(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 n + k disks. Thus the problem of packing equal disks is the special case of our problem with n=h=0 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 h = 0. Our main algorithmic contribution is an algorithm that solves the repacking problem in time (h+k) ^ {O (h+k) } |I|^ {O (1) } (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.