Line Segment Disk Cover

Abstract

In this paper, we consider the following variations of Line Segment Disk Cover (LSDC) problem.

• $\text {LSDC-H:}$ In this version of LSDC problem, we are given a set $\mathcal{S}=\{s_1, s_2, \ldots , s_n\}$ of n horizontal line segments of arbitrary length and an integer $k({\ge }1)$. Our aim is to cover all segments in $\mathcal{S}$ with k disks of minimum radius centered at arbitrary points in the plane.

• $\text {LSDC-A:}$ In this version of LSDC problem, we are given a set $\mathcal{S}=\{s_1, s_2, \ldots , s_n\}$ of n line segments of arbitrary length with arbitrary orientation and an integer $k({\ge }1)$. Our aim is to cover all segments in $\mathcal{S}$ with k disks of minimum radius centered at arbitrary points in the plane.

• $\text {LSDC-D:}$ In the discrete version of LSDC problem, we are given a set $\mathcal{S}=\{s_1, s_2, \ldots , s_n\}$ of n line segments of arbitrary length with arbitrary orientation and a set $\mathcal{D}=\{d_1, d_2, \ldots , d_m\}$ of m disks of unit radius. Our aim is to cover all segments in $\mathcal{S}$ with minimum number of disks in $\mathcal{D}$ i.e. $\mathcal{S}\subset \bigcup \limits _{_{d\in D'}}d$, where $D'\subseteq \mathcal{D}$ is of minimum cardinality.

For LSDC-H and LSDC-A problems, we propose $(1+\epsilon )$-factor approximation algorithms, which run in $O({(\lceil \frac{\pi }{\delta ^2}\rceil )}^{k}n(|{\log r_{opt}}|+\log \lceil \frac{1}{\rho }\rceil ))$ time and $O({(\lceil \frac{\pi }{\delta ^2}\rceil )}^{k}n\log n(|{\log r_{opt}}|+\log \lceil \frac{1}{\rho }\rceil ))$ time respectively, where $r_{opt}$ is the minimum radius of k disks which cover all segments in $\mathcal{S}$, and $\delta >0$, $\rho >0$ and $\epsilon >0$ are fixed constants such that $\epsilon \ge (\delta +\delta \rho +\rho )$. For LSDC-D problem, we propose a $(1+\epsilon )$-factor approximation algorithm (PTAS), which runs in $O(m^{2(\frac{8\sqrt{2}}{\epsilon })^2+3}+m^2n)$ time, and a $(9+\epsilon )$-factor approximation algorithm, which runs in $O(m^{(5+\frac{18}{\epsilon })}\log m+m^2n)$ time, where a constant $\epsilon >0$.