Split Packing: An Algorithm for Packing Circles with up to Critical Density

In the classic circle packing problem, one asks whether a given set of circles can be packed into the unit square. This problem is known to be NP-hard. In this thesis, we present a new sufficient condition using only the circles’ combined area: It is possible to pack any circle instance with a combined area of up to ≈53.90% of the square’s area. This area condition is tight, in the sense that for any larger percentage, there are instances which cannot be packed. We call the ratio between the largest area that can always be packed and the area of the container the critical density. Similar results have long been known for squares, but to the best of our knowledge, this thesis gives the first results of this type for circular objects. Our proof is constructive: We describe a versatile, divide-and-conquer-based algorithm for packing circles and other objects into various container shapes with up to critical density. It employs an elegant subdivision scheme which recursively splits the circles into two groups and then packs these into subcontainers. We call the algorithm Split Packing. Beside realizing all packings below the critical density bound, there is a second perspective on this algorithm: It can be used as a constant-factor approximation algorithm when looking for the smallest container in which a given set of circles can be packed, due to its polynomial runtime. In this thesis, we demonstrate that the Split Packing approach is applicable to a large number of different packing problems. Beside for square containers, we are able to apply the algorithm to pack circles with critical density into right and obtuse triangles, equilateral triangles, isosceles triangles whose longest side is their base, and rectangles with an aspect ratio of more than≈1.57 : 1. Additionally, we show that the same algorithm can be used to pack objects other than circles: It can pack octagons into squares and isosceles right triangles with critical density, and if allowed to rotate the objects, it can even be used to pack squares into these containers with critical density. Aside from the obtained results, we believe that the ideas behind Split Packing are interesting and elegant on their own. We see many opportunities to apply these techniques in the context of other packing and covering problems. A browser-based, interactive visualization of the Split Packing approach and other related material can be found at https://morr.cc/split-packing/.