Here is a modification of the weighted interval scheduling problem.
Given $n$ intervals by their end points $s_i$ and $t_i$ for $1 ≤ i ≤ n$, I want to minimize number of intervals so that every interval has a nonempty intersection with one of the selected intervals. In other words, select $S ⊆ {1,..., n}$ such that for each $1 ≤ j ≤ n$, interval $j$ intersects with some interval $i ∈ S$. We say that the $j^{th}$ interval $[s_j, t_j]$ overlaps with $i^{th}$ interval $[s_i, t_i]$ if there exists a point $x$ such that $x$ belongs to the $i^{th}$ interval and the $j^{th}$ interval. A point $x$ belongs to an interval $[s, t]$ if $s ≤ x ≤ t$. Assume that $s_i ’s$ and $t_i's$ are distinct.
What I'm confused on is what this problem is even asking. It says minimize the number of intervals so that every interval has a non-empty intersection with a selected interval. So I get that means I want to find an interval that overlaps with another interval. But am I trying to find a sequence that avoids this as much as possible? In other words, am I just trying to set it up so that I maximize number of intervals while also minimizing overlapping intervals or am I trying to do something different?