Chips on wafers (Extended abstract)

A set of rectangles 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 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 of rectangles and a real constant produces a grid packing of whose area is at most times larger than an optimal packing in polynomial time. If 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 rectangles, and given a region grid pack as many rectangles as possible within . Apart from the approximation algorithms we present several hardness results