Hardness in Perfect Rectangle Packing Problems

Rectangle Packing is considered a difficult NP-problem which has several application areas such as integrated circuit systems, and pallet packing. Generally, Rectangle Packing Problems consist of a set of rectangles for which a configuration must be found, the intention is to minimise the surface-area of the configurations bounding box. Existing research has predominantly investigated algorithms to solve these problems as quickly as possible. Conversely, this study focuses on predicting the hardness of Perfect Rectangle Packing Problems, as the difficulty has been observed to vary in degrees. This may enable application areas to avoid the hardest Rectangle Packing Problems. This study proposes the use of the ES-attribute of a rectangle set, which is based on the number of shared sides it has, for predicting the difficulty of rectangle packing problems in terms of solution density. To support this proposition an algorithm was designed to generate rectangle-sets uniformly, subsequently a sizable set of rectangle-sets was generated. The number of solutions was calculated for each rectangle set; any configuration with zero empty-space is considered a solution, and sets with no perfect configuration were discarded. Accordingly, the total number of solutions is considered as a measure of difficulty. Subsequently, the data suggests an exponential correlation between the ES-attribute and the average difficulty of a Perfect Rectangle Packing Problem. The strength of this correlation is evaluated in relation to the cardinality of rectangle-sets and the ratio of bounding boxes. The results suggest the correlation is able to estimate the average number of solutions for a great amount of rectangle-set, although, it is inaccurate for single instances. However, instances with a large number of solutions occur more often at large ES-values, whereas at low ES-values only instances with an extremely low number of solutions exist. Thus, for an easy Perfect Rectangle Packing Problem a large ES-value is required, however, a large ES-value does not guarantee an easy problem instance.