I haven't been able to solve this problem for a long time (10 months), so I would greatly appreciate anyone can solve it or can provide hints.

For $$$ N, K \le 2*10^3 $$$ subtask we can use a straightforward DP, but I have no idea for DP optimization or Greedy algorithm for $$$ N, K \le 2*10^5 $$$ full task.

Examples:

6 2
2 -1 3 -4 -2 6

Answer: segments [1, 3] and [6, 6] with sum 10. No need to output which segments are used, just the sum itself.