Optimal covering of line subsegments using a given set of disks Is there a way of picking a minimal set of disks that's still covering the same line subsegments as all the disks together? Any help or references highly appreciated. Below is just an illustrative picture, because I actually have 10.000 disks.

 A: This will be a high-level suggestion, and definitely not optimal.
First, execute a sweepline algorithm to detect all the points of intersections between
segments and circles.
Then for each segment, run along it and discard portions not covered by any disk. Now
you are left with subsegments, each of which is covered by one or more disks.
For each disk, record which subsegments it covers. 
Discard a disk if it covers no subsegment. Now the suboptimal part: if all of the subsegments a particular disk covers are
covered by more than one disk, discard that disk, and repeat.
This is a mindless discarding and would not in general achieve the minimal cover.
If you really need the minimal cover, you'll have to proceed analogously to 
this paper, as the problem is almost certainly NP-hard:

Alt, Helmut, Esther M. Arkin, Hervé Brönnimann, Jeff Erickson, Sándor P. Fekete, Christian Knauer, Jonathan Lenchner, Joseph SB Mitchell, and Kim Whittlesey. "Minimum-cost coverage of point sets by disks." In Proceedings of the 22nd annual symposium on Computational geometry, pp. 449-458. ACM, 2006. 
  arXiv preprint cs/0604008.

A: You can formulate this as a set covering problem.  For each circle $j$, define a binary variable $x_j$ that indicates whether circle $j$ is selected.  Let $C_i$ be the set of circles that intersect line segment $i$.  The problem is to minimize $\sum\limits_j x_j$ subject to
$\sum\limits_{j \in C_i} x_j \ge 1$
for each line segment $i$ with $|C_i|>0$.
