How to find optimal partition subsets
First, let me be sure u r aware of SET PACKING &amp; SET PARTITIONING Problems  

https://slideplayer.com/slide/4514539/

Then, I think ur problem is much easier avoiding the overlapping &amp; the complete coverage constraints.

Going simple, here r some heuristics (think as u solving for max/optimal accepting 30%-40%"deviation from optimality) 

1-Pick the max. length partition, then in the next 9th steps choose bet the global max of remaining partitions &amp; the max of those starting after what u have reached so far.... I think that's O(10*5000) let's call it $ O(k n) $

2-Do something similar to $Bucket$ $Sorting$ 

Divide the partition to 10 smaller partitions(put an overlapping sub partition in its starting point bucket) , then select the max length one in each bucket. Still the same order complexity $ O(k*n) $

Finally, maybe a little bit complicated, but if u want a Theoretical abstraction of ur problem I've just watched this about the $maximum$ $K$ $coverage$ problem and the $Influence$ $maximization$ problem.

If u looked to the interval [0,1] as say M smaller/unit sub intervals being ur target population, and viewed each sub partition as connected to/infleuncing the sub intervals(units) it covers. Then u want to select $K$ (10 in ur case) subpartitions that gives u maximum coverage/influence

https://youtu.be/GqKbx3LCQqE