Exact solutions of rectangular partitions via integer programming

摘要

Given a rectangle R in the plane and a finite set P of points in its interior, consider the partitions of the surface of R into smaller rectangles. A partition is feasible with respect to P if each point in P lie on the boundary of some rectangle of the partition. The length of a partition is computed as the sum of the lengths of the line segments defining the boundary of its rectangles. The goal is to find a feasible partition with minimum length. This problem, denoted by RGP, belongs to NP-hard and has application in VLSI design. In this paper we investigate how to obtain exact solutions for the RGP. We introduce two different Integer Programming formulations and carry out a theoretical study to evaluate and compare the strength of their bounds. Computational experiments are reported for Branch-and-Cut and Branch-and Price algorithms we have implemented for the first and the second formulation, respectively. Randomly generated instances with |P| ≤ 200 are solved exactly. The tests indicate that the size of the instances 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.