Minimum Rectilinear Polygons for Given Angle Sequences

TL;DR

This paper investigates the computational complexity of constructing rectilinear polygons with minimal perimeter, area, or bounding box area based on given angle sequences, proving NP-hardness in general but providing efficient solutions for certain cases.

Contribution

It establishes NP-hardness for minimizing perimeter, area, and bounding box area of rectilinear polygons, and offers efficient algorithms for x-monotone and xy-monotone cases.

Findings

NP-hardness of perimeter, area, and bounding box minimization in general cases

Efficient algorithms for x-monotone and xy-monotone rectilinear polygons

Resolution of an open problem on NP-hardness for bounding box area minimization

Abstract

A rectilinear polygon is a polygon whose edges are axis-aligned. Walking counterclockwise on the boundary of such a polygon yields a sequence of left turns and right turns. The number of left turns always equals the number of right turns plus 4. It is known that any such sequence can be realized by a rectilinear polygon. In this paper, we consider the problem of finding realizations that minimize the perimeter or the area of the polygon or the area of the bounding box of the polygon. We show that all three problems are NP-hard in general. This answers an open question of Patrignani [CGTA 2001], who showed that it is NP-hard to minimize the area of the bounding box of an orthogonal drawing of a given planar graph. We also show that realizing polylines with minimum bounding box area is NP-hard. Then we consider the special cases of $x$-monotone and $xy$-monotone rectilinear polygons. For these, we can optimize the three objectives efficiently.