Improved PTAs for the Unit-Height Rectangle Packing Problem: A New Dynamic Programming Procedure

Abstract

We consider the following problem: given a set of weighted axis-parallel unit-height closed rectangles on the plane find a maximum weight subset in which no two rectangles intersect. This problem is equivalent to the maximum weight independent set problem in the intersection graph of rectangles of unit height. The previously known polynomial time approximation scheme described in Agarwal et al [1] uses a shifting technique of Hochbaum and Maass [6] in combination with a dynamic programming procedure. Its running time and memory consumption are both ${\rm O}\left( {{n^{{2k - 1}}}} \right)$ with respect to approximation factor of $\left( {1 - 1/k} \right)$, where k is a fixed constant $(k \geqslant 2)$ and n is number of rectangles in the input. We describe a new dynamic programming procedure which enables us to reduce the time and memory requirements of that PTAS to $O({k^2}{n^k}\log n)$ and $O({n^{{k - 1}}})$ respectively.