Codeforces Round 1040 (Div. 2) Problem C
Observations
Given n ranges, the problem involves maximally covering the range [1, 2n] with those ranges, while jointly minimizing cycles between the endpoints of those ranges/ undirected edges. [1, 2n] will be maximally covered by taking all ranges, but this does not necessarily avoid cycles. The important observation is that all cycles have a minimum and maximum point, e.g. 1 and 4 in S = {(1, 2), (2, 4), (1, 4)}. There are always two "paths" to get from the minimum to the maximum. In the case of S, {(1, 2), (2, 4)} or {(1, 4)}. Getting from the minimum to the maximum while avoiding a cycle is then just a matter of take one path and only one path.
Take ranges/ undirected edges greedily. Only take an edge (a, b) if b is not currently reachable from a, i.e. the other "path" has not already been taken