Chips on wafers

A set of rectangles S is said to be grid packed if there exists a rectangular grid (not necessarily regular) such that every rectangle lies in the grid and there is at most one rectangle of S in each cell. The area of a grid packing is the area of a minimal bounding box that contains all the rectangles in the grid packing. We present an approximation algorithm that given a set S of rectangles and a real constant epsilon > 0 produces a grid packing of S whose area is at most (I + epsilon) times larger than an optimal packing in polynomial time. If epsilon is chosen large enough the running time of the algorithm will be linear. We also study several interesting variants, for example the smallest area grid packing containing at least k less than or equal to n rectangles, and given a region A grid pack as many rectangles as possible within A. Apart from the approximation algorithms we present several hardness results.