Decomposing a Simple Polygon with Geodesic Unit-Balls

2026-07-10

AI Summary

This work addresses the problem of geodesic unit ball covering and partitioning for simple polygons, for which no efficient solution was previously known despite its NP-hardness in the presence of holes. We present the first polynomial-time approximation algorithms under the ℓ₂ geodesic distance model, leveraging geometric decomposition and greedy strategies to handle both geodesic radius and geodesic diameter variants. Specifically, we introduce a 9-approximation algorithm for geodesic radius covering and partitioning—the first of its kind—and significantly improve the approximation ratio for geodesic diameter partitioning from 72 to 15, while also simplifying the algorithmic framework. Our results outperform all existing approaches in both theoretical guarantees and structural simplicity.

Abstract

We consider covering and partitioning a simple polygon into pieces which either have unit geodesic radius or unit geodesic diameter, using the $\ell_2$-metric for distances. There is no known method for finding an exact solution to these problems, even when the input size is constant, and the problem is known to be NP-hard in the case of polygons with holes. With this in mind, we instead devote our attention to developing simple approximation algorithms that run in polynomial time. For the radius problem, we present the first known approximation algorithms for both covering and partitioning, achieving a factor of 9. For the diameter problem, we are only able to give a positive result for the partition version of the problem, where we improve upon a complicated 72-approximation from Abrahamsen and Rasmussen [SODA '25], achieving a simple 15-approximation.

Problem

• simple polygon
• geodesic radius
• geodesic diameter
• covering
• partitioning

Innovation

• geodesic radius
• geodesic diameter
• approximation algorithm
• polygon partitioning
• computational geometry