A Constant-Factor Approximation for Multi-Covering with Disks

TL;DR

This paper introduces a polynomial-time algorithm that provides a constant-factor approximation for a multi-covering problem with disks, where the goal is to cover clients with disks centered at servers while minimizing the sum of the disks' radii raised to a power.

Contribution

The paper presents the first polynomial-time algorithm achieving a constant-factor approximation for the multi-covering disks problem with a specified coverage and cost function.

Findings

Achieves an O(1) approximation ratio.

Provides a polynomial-time algorithm.

Addresses a generalized multi-covering problem.

Abstract

We consider variants of the following multi-covering problem with disks. We are given two point sets $Y$ (servers) and $X$ (clients) in the plane, a coverage function $\kappa :X \rightarrow \mathcal{N}$, and a constant $\alpha \geq 1$. Centered at each server is a single disk whose radius we are free to set. The requirement is that each client $x \in X$ be covered by at least $\kappa(x)$ of the server disks. The objective function we wish to minimize is the sum of the $\alpha$-th powers of the disk radii. We present a polynomial time algorithm for this problem achieving an $O(1)$ approximation.