Efficient constant factor approximation algorithms for stabbing line segments with equal disks
An NP-hard problem is considered of intersecting a given set of $n$ straight line segments on the plane with the smallest cardinality set of disks of fixed radii $r>0,$ where the set of segments forms a straight line drawing $G=(V,E)$ of a planar graph without proper edge crossings. To the best of our knowledge, related work only tackles a setting where $E$ consists of (generally, properly overlapping) axis-parallel segments, resulting in an $O(n\log n)$-time and $O(n\log n)$-space 8-approximation algorithm. Exploiting tough connection of the problem with the geometric Hitting Set problem, an $\left(50+52\sqrt{\frac{12}{13}}+ν\right)$-approximate $O\left(n^4\log n\right)$-time and $O\left(n^2\log n\right)$-space algorithm is devised based on the modified Agarwal-Pan algorithm, which uses epsilon nets. More accurate $(34+24\sqrt{2}+ν)$- and $\left(\frac{144}{5}+32\sqrt{\frac{3}{5}}+ν\right)$-approxi\-mate algorithms are also proposed for cases where $G$ is any subgraph of either a generalized outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where $ν>0$ is an arbitrarily small constant.