On Guillotine Separability of Squares and Rectangles

Guillotine separability of rectangles has recently gained prominence in combinatorial optimization, computational geometry, and combinatorics. Consider a given large stock unit (say glass or wood) and we need to cut out a set of required rectangles from it. Many cutting technologies allow only end-to-end cuts called guillotine cuts. Guillotine cuts occur in stages. Each stage consists of either only vertical cuts or only horizontal cuts. In k-stage packing, the number of cuts to obtain each rectangle from the initial packing is at most k (plus an additional trimming step to separate the rectangle itself from a waste area). Pach and Tardos [Pach and Tardos, 2000] studied the following question: Given a set of n axis-parallel rectangles (in the weighted case, each rectangle has an associated weight), cut out as many rectangles (resp. weight) as possible using a sequence of guillotine cuts. They provide a guillotine cutting sequence that recovers 1/(2 log n)-fraction of rectangles (resp. weights). Abed et al. [Fidaa Abed et al., 2015] claimed that a guillotine cutting sequence can recover a constant fraction for axis-parallel squares. They also conjectured that for any set of rectangles, there exists a sequence of axis-parallel guillotine cuts that recovers a constant fraction of rectangles. This conjecture, if true, would yield a combinatorial O(1)-approximation for Maximum Independent Set of Rectangles (MISR), a long-standing open problem. We show the conjecture is not true, if we only allow o(log log n) stages (resp. o(log n/log log n)-stages for the weighted case). On the positive side, we show a simple O(n log n)-time 2-stage cut sequence that recovers 1/(1+log n)-fraction of rectangles. We improve the extraction of squares by showing that 1/40-fraction (resp. 1/160 in the weighted case) of squares can be recovered using guillotine cuts. We also show O(1)-fraction of rectangles, even in the weighted case, can be recovered for many special cases of rectangles, e.g. fat (bounded width/height), δ-large (large in one of the dimensions), etc. We show that this implies O(1)-factor approximation for Maximum Weighted Independent Set of Rectangles, the weighted version of MISR, for these classes of rectangles.