graph - How to find Minimum Directed Cycle (minimum total weight)?

You can use Floyd-Warshall algorithm here.

The Floyd-Warshall algorithm finds shortest path between all pairs of vertices.

The algorithm is then very simple, go over all pairs (u,v), and find the pair that minimized dist(u,v)+dist(v,u), since this pair indicates on a cycle from u to u with weight dist(u,v)+dist(v,u). If the graph also allows self-loops (an edge (u,u)) , you will also need to check them alone, because those cycles (and only them) were not checked by the algorithm.

pseudo code:

run Floyd Warshall on the graph
min <- infinity
vertex <- None
for each pair of vertices u,v
    if (dist(u,v) + dist(v,u) < min):
           min <- dist(u,v) + dist(v,u)
           pair <- (u,v)
return path(u,v) + path(v,u)

path(u,v) + path(v,u) is actually the path found from u to v and then from v to u, which is a cycle.

The algorithm run time is O(n^3), since floyd-warshall is the bottle neck, since the loop takes O(n^2) time.

I think correctness in here is trivial, but let me know if you disagree with me and I'll try to explain it better.

Is my algorithm correct?

No. Let me give a counter example. Imagine you start DFS from u, there are two paths p1 and p2 from u to v and 1 path p3 from v back to u, p1 is shorter than p2.

Assume you start by taking the p2 path to v, and walk back to u by path p3. One cycle found but apparently it's not minimum. Then you continue exploring u by taking the p1 path, but since v is fully explored, the DFS ends without finding the minimum cycle.