Decomposing Polygons into Fat Components

TL;DR

This paper investigates polygon decomposition into fat components, providing a polynomial-time algorithm for simple polygons and proving NP-hardness for polygons with holes regarding minimal fat partitions.

Contribution

It introduces a polynomial-time algorithm for decomposing simple polygons into fat components and proves NP-hardness for polygons with holes for minimal fat partitions.

Findings

Polynomial-time algorithm for simple polygons

NP-hardness for polygons with holes

Optimal fat partition parameters computed efficiently

Abstract

We study the problem of decomposing (i.e. partitioning and covering) polygons into components that are $\alpha$-fat, which means that the aspect ratio of each subpolygon is at most $\alpha$. We consider decompositions without Steiner points. We present a polynomial-time algorithm for simple polygons that finds the minimum $\alpha$ such that an $\alpha$-fat partition exists. Furthermore, we show that finding an $\alpha$-fat partition or covering with minimum cardinality is NP-hard for polygons with holes.