Exact Solutions of Rectangular Partitions via Integer Programming

We are given a rectangle R in the plane and a nite set P of n points in the interior of R. A rectangular partition of R is a partition of the surface of R into smaller rectangles. A rectangular partition is said to be feasible with respect to P if no points in P lie in the strict interior of any rectangle of the partition. The length of a rectangular partition is computed as the sum of the lengths of the straight line segments that deene the boundary of its rectangles. The goal is to nd a (feasible) rectangular partition whose length is minimized. This problem, denoted here by RGP, is NP-hard and has application in VLSI design. Several approximation algorithms have been proposed in the literature to solve it. In this paper we investigate how to obtain exact solutions for the RGP. We introduce two diierent Integer Programming formulations. To test these formulations computationally we have implemented a Branch-and-Cut algorithm and a Branch-and-Price algorithm for the rst and the second formulation, respectively. Comparisons between the two formulations are made. The computational results with a randomly generated set of instances show that it is possible to solve exactly medium-sized instances of the RGP. Moreover, we show that the size of the instances that can be solved with our algorithms decrease by an order of magnitude in the absence of corectilinear points in P, a special case of RGP whose complexity is still open. Finally we also have implemented the best approximation algorithm known for RGP and we show that it usually produces solutions about 10% oo the optimum.