Approximation of Two-Dimensional Rectangle Packing

1-d bin packing and 2-d bin packing are classic NP-complete problems that are motivated by industrial applications of stock cutting, data storage, etc. In this paper, we look at a slightly more general problem that is primarily motivated by VLSI design. In VLSI design, one wants to pack a given number of rectangles (which might be RAM chips, transistors, etc.) into a wafer of minimum area to reduce the size of the chip. This is the 2-d rectangular packing problem. The optimization problem is “Given a set of N rectangles, minimize the area of a bounding box that contains all the N rectangles, without any rotations and such that no two rectangles overlap.” The corresponding decision problem is “Given a set of N rectangles and a box of length L and height H, can all the N rectangles be packed in the box?” By a reduction from the 2-d bin packing, it can be easily shown that the problem is NP-hard. Also it can be easily seen that the problem is in NP, and hence it is NP-complete. Though a lot of research has gone into 1-d bin packing and 2-d bin packing [CGJ84],[GJ79],[CLR90], the rectangle packing problem is relatively less researched. Coffman, et.al. [CGJ84], in addition to discussing the bin-packing problems and its variations, present a brief survey of a variation closer to the rectangle packing problem. In this variation of the problem, called strip-packing, they try to minimize the area after fixing the width of the bounding box. This problem arises in a variety of industrial settings where the “raw” material comes in the form of rolls and from this we may wish to cut rectangular patterns for various purposes. Due to the economic importance of efficient stock cutting, this variant has been extensively studied and various heuristics and approximation algorithms exist for that. References to papers studying this problem can be found in [CGJ84]. In each of these algorithms, the rectangles are first sorted according to a sorting rule, like decreasing width (DW), increasing height (IH), etc. and then packed according to some packing rule, like the BOTTOM-LEFT (BL) rule where the sorted rectangles are packed in turn, each item being placed as close to the bottom of the strip as it will fit and then as far to the left as it can be placed at that bottom-most level. For this simple approach, the BLDW algorithm, where we use the BOTTOM-LEFT algorithm after ordering the blocks by decreasing width gives us the best approximation ratio of 3. Another similar set of algorithms are based on a different type of packing rule called level algorithms. The rectangles are all sorted by decreasing heights. Two packing rules called NEXT FIT and FIRST FIT give good results with the asymptotic worst case ratios for the two being 2 and 1.7 respectively. These algorithms require guillotine cuts, i.e. edge-to-edge cuts of the strip parallel to the bottom of the strip. Without this constraint, the best algorithm based on similar ideas is an algorithm by Baker, et. al. [BBK81] which gives an asymptotic worst case ratio of 1.25. Klietman and Krieger [KK75] address the rectangle packing problem for the special case when all rectangles are squares. They consider a collection of squares whose total area add up to unity. They show that a bounding box of 2= p 3 by p 2 is a unique rectangle that will suffice. Hochbaum and Maass [HM85] propose a shifting strategy to address the special case of square packing. They consider another variation where they try to pack a maximal number of k k squares (for a natural number k) into an area that is given by n squares of unit size on a rectilinear grid. For any natural number l 1, the approximation ratio is (1+1=l) and the run-time is O(k2l2nl). They also propose ways to obtain approximations to packing problems in higher dimensions, with arbitrary orientations and with objects other than squares. Hwang, Kao and Horng [HKH94] present a genetic algorithm to solve the rectangle packing problem where 90o rotation of blocks is allowed and square packing is preferred. They also address the bin packing and strip