An Approximation Algorithm for the H-Prize-Collecting Power Cover Problem

Abstract

We are given a set U of user points, a set $\mathcal{S}$ of sensors in a d-dimensional space $\mathbb {R}^d$ and a lower bound H. Each user point $u\in U$ has a profit h(u) and a penalty cost $\pi (u)$. Each sensor $s\in \mathcal{S}$ can adjust its power, and the cover range of sensors with power p(s) is a d-dimensional ball of radius r(s), where $p(s)= r(s)^{\alpha }$ and $\alpha \ge 1$ is a constant. The goal of the H-prize-collecting power cover problem is to determine a power assignment such that the total profit of covered user points is at least H and the total power of sensors plus the total penalty cost of uncovered user points is minimized. First, we proved that this problem is NP-hard even when $\alpha =1$, and $d=1$ and $\pi (u)=0$ for any $u\in U$. Then, by utilizing primal-dual and Lagrangian relaxation techniques, we present a $(4\cdot 3^{\alpha -1}+ \epsilon )$-approximation algorithm for any desired accuracy $\epsilon >0$.