Geometric Bin Packing Beyond Convexity: Approximation Schemes without Augmentation

Abstract

We study the geometric bin packing problem for simple polygonal regions, i.e., non-degenerate, connected, rational regions, in the minimum number of unit-square bins, allowing only translations. We even consider packing general non-convex objects, possibly with holes, where we can then pack smaller objects. Previous schemes cover squares (MOR'06), circles and other $L_p$-balls (Algorithmica'16), sufficiently skewed rectangles (Algorithmica'23), and convex fat objects with a constant number of distinct shapes under resource augmentation (arXiv'24, WAOA'23). The result for circles and other $L_p$-balls needs bin augmentation to obtain rational packing realizations. For restricted classes of $L$-shaped polygons, Gerlach et al. (SoCG'26) give constant worst-case guarantees, while for general non-convex objects, bounding-box reductions give just approximation factors that depend on fatness. Against this background, we present a general framework that provides asymptotic polynomial-time approximation schemes (APTAS) for bin packing of any fixed class of simple polygonal regions satisfying just two assumptions. First, covering any object by the grid cells intersecting its interior in any sufficiently fine square grid, relative to the object's bounding box, increases its area by at most a small fraction. Second, the class can be partitioned into a constant number of sequences such that each object in a sequence can be translated to fit inside every preceding object. In particular, certain classes of star-shaped polygons and more general polygonal regions, including some with simple holes, satisfy both assumptions. In addition, our framework does not require a bound on the objects' boundary complexity. We emphasize that our results require no bin augmentation whatsoever and yield explicit, exact rational packings of polynomial encoding length for all input objects.