Cutting Polygons into Small Pieces with Chords: Laser-Based Localization

TL;DR

This paper investigates algorithms and complexity for partitioning polygons into small pieces using chords, motivated by laser-based indoor localization, considering various size measures and optimization objectives.

Contribution

It introduces new approximation algorithms and hardness results for polygon partitioning problems with multiple size measures and constraints.

Findings

Provided approximation algorithms for various polygon partitioning variants.

Established hardness results for polygons with holes.

Analyzed different size measures like area, diameter, and inscribed circle radius.

Abstract

Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different objectives, either minimizing the maximum size of a piece for a given number of chords, or minimizing the number of chords that achieve a given size threshold for the pieces. We give hardness results for polygons with holes and approximation algorithms for multiple variants of the problem.