Find minimum number of points which intersect overlapping arcs
A little abstraction would help both reasoning about and solving the problem efficiently, I think.
For every point $x$ on the circle, there is some set $S_x$ of pairwise overlapping arcs that intersect this point. Now, if there is some other point $y$ such that $S_x \subseteq S_y$, you can always use $y$ instead of $x$ in a solution to your problem. We can therefore focus on a set of representative points $x_1,\ldots,x_m$ such that each set $S_{x_1},\ldots,S_{x_m}$ is maximal. There are at most $n$ such points (the proof of which I will leave as an excercise) and you should be well able to find this set in $O(n^2)$ time.
Now, if the arcs don't wrap all around the circle, i.e. there is some point on the circle that doesn't touch any arcs, then you can look at the arcs as intervals on a straight line. Given the right infrastructure (you should sort the arcs both by starting and ending angle, again easily doable in $O(n^2)$ time) you can solve this problem greedily in $O(n)$ time (cf. this post).
Finally, to solve the problem, you do as follows: For each $1\leq i\leq m$, you ask the following question: &quot;If i use $x_i$, how many other points must I use in order to cover the arcs that are not in $S_{x_i}$?&quot; These arcs naturally do not wrap around the circle. You can therefore calculate this in linear time, and return the minimum set of points over all tries. Since $m\leq n$, this will only take $n^2$ time in total.