A Divide and Conquer Approximation Algorithm for Partitioning Rectangles

Given a rectangle $R$ with area $A$ and a set of areas $L=\{A_1,\mathrm{.}\mathrm{.}\mathrm{.},A_n\}$ with ${\sum }_{i=1}^n{A}_i=A$, we consider the problem of partitioning $R$ into $n$ sub-regions $R_1,\mathrm{.}\mathrm{.}\mathrm{.},R_n$ with areas $A_1,\mathrm{.}\mathrm{.}\mathrm{.},A_n$ in a way that the total perimeter of all sub-regions is minimized. The goal is to create square-like sub-regions, which are often more desired. We propose an efficient $1.203$–approximation algorithm for this problem based on a divide and conquer scheme that runs in $\mathcal{O}(n^2)$ time. For the special case when the aspect ratios of all rectangles are bounded from above by 3, the approximation factor is $2\mathrm{/}\sqrt{3}\le 1.1548$. We also present a modified version of out algorithm as a heuristic that achieves better average and best run times.