A Divide and Conquer Approximation Algorithm for Partitioning Rectangles

TL;DR

This paper introduces a divide and conquer approximation algorithm for partitioning a rectangle into sub-regions with specified areas, aiming to minimize total perimeter and favor square-like shapes, with proven approximation bounds.

Contribution

It presents a novel $1.203$--approximation algorithm for rectangle partitioning with a divide and conquer approach, including a specialized version for bounded aspect ratios.

Findings

Achieves a $1.203$--approximation factor in $O(n^2)$ time.

Special case with aspect ratios ≤ 3 has an approximation factor of ≤ 1.1548.

Modified heuristic improves average and best run times.

Abstract

Given a rectangle $R$ with area $A$ and a set of areas $L=\{A_1,...,A_n\}$ with $\sum_{i=1}^n A_i = A$, we consider the problem of partitioning $R$ into $n$ sub-regions $R_1,...,R_n$ with areas $A_1,...,A_n$ 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 $2/\sqrt{3} \leq 1.1548$. We also present a modified version of out algorithm as a heuristic that achieves better average and best run times.