# Range Assignment of Base-Stations Maximizing Coverage Area without Interference

## TL;DR

This paper investigates the problem of assigning non-overlapping geometric objects to maximize total coverage area, proving NP-hardness, analyzing existing algorithms, and proposing a PTAS with extendable approximation guarantees.

## Contribution

The paper establishes NP-hardness for area maximization, analyzes Eppstein's perimeter algorithm as a 2-approximation, and introduces a PTAS for the problem, extendable to higher dimensions.

## Key findings

- NP-hardness of area maximization with non-overlapping objects
- Eppstein's perimeter algorithm yields a 2-approximation for area maximization
- Proposed PTAS provides near-optimal solutions, extendable to higher dimensions

## Abstract

We study the problem of assigning non-overlapping geometric objects centered at a given set of points such that the sum of area covered by them is maximized. If the points are placed on a straight-line and the objects are disks, then the problem is solvable in polynomial time. However, we show that the problem is NP-hard even for simplest objects like disks or squares in ${\mathbb{R}}^2$. Eppstein [CCCG, pages 260--265, 2016] proposed a polynomial time algorithm for maximizing the sum of radii (or perimeter) of non-overlapping balls or disks when the points are arbitrarily placed on a plane. We show that Eppstein's algorithm for maximizing sum of perimeter of the disks in ${\mathbb{R}}^2$ gives a $2$-approximation solution for the sum of area maximization problem. We propose a PTAS for our problem. These approximation results are extendible to higher dimensions. All these approximation results hold for the area maximization problem by regular convex polygons with even number of edges centered at the given points.