Approximating the Packing of Unit Disks into Simple Containers∗

We study the following problems: Given a triangle or parallelogram, how many unit-disks can be packed without overlap into it? It is not known whether these problems are NP-hard or in NP. We give the first results on their approximability and therefore a better understanding of their complexity. In the case that all inner angles are bounded from below by a constant, we give a PTAS for each problem. For the case of arbitrarily small inner angles, we give an algorithm with constant-factor approximation and polynomial running time for triangles.