Orthogonal dissection into few rectangles

TL;DR

This paper presents a polynomial-time algorithm for dissecting polygons into the fewest axis-aligned rectangles, utilizing the Dehn invariant, and extends to minimal-edge dissections and approximations with rotations.

Contribution

It introduces a novel polynomial-time method for minimal rectangle dissection based on the Dehn invariant, including extensions for minimal-edge polygons and approximate solutions with rotations.

Findings

Dissects polygons into minimal rectangles using Dehn invariant.

Can also minimize edges in polygon dissection.

Provides a 2-approximation when rotations are allowed.

Abstract

We describe a polynomial time algorithm that takes as input a polygon with axis-parallel sides but irrational vertex coordinates, and outputs a set of as few rectangles as possible into which it can be dissected by axis-parallel cuts and translations. The number of rectangles is the rank of the Dehn invariant of the polygon. The same method can also be used to dissect an axis-parallel polygon into a simple polygon with the minimum possible number of edges. When rotations or reflections are allowed, we can approximate the minimum number of rectangles to within a factor of two.