I'm solving an optimization problem involving several sets of ranges: there are n sets of exclusion ranges, I need to find a set of segments, that has the least overlap with the exclusion ranges, while keeping the distance between individual segments below a certain threshold and the segments' length above a certain minimum. To better visualize the problem, I've attached a picture.
The sets of exclusion ranges have varying exclusion priority, so that avoiding x amount of overlap with set 1 is, say, twice more important than avoiding x amount of overlap with set 2. A trivial, non-optimized solution would then be to have a single segment that has no breaks. I've started solving the problem by coding a simple generator of such exclusion ranges in Python and having the program draw a "score" function - sample a bunch of points for what sets they belong to, give each point a rating for what sets it lies in - 3 points for set 1, 2 for set 2, 1 for set 3. I'm attaching the output of the function to better explain myself.
I figure that to solve the problem at hand I will attempt to minimize the sum of integrals of this function across the segments. The question is how to move the segments, while still observing the rules of their placement (no farther from one another that x, no shorter than y). I've an idea of how to do that, but I fear it will not only reach a suboptimal result, but also require a lot of computations, so I ask you for either a good idea of how to approach the problem, or, better yet, of some articles or books on similar topics.
Thank you!
