How can I modify Floyd-Warshall so that for any pair of vertices, the top-K shortest path can found with minimum cost? For example, the shortest path with k = 3 intermediate vertices from v1 to v5 is v1 → v3 → v2 → v3 → v5 for a total cost of w(v1, v3) + w(v3, v2) + w(v2, v3) + w(v3, v5) = 4 + 1 + 1 + 2 = 8
I am expecting, for example the path from v1 to v1 is 6(instead of 0) for k =3 and the path from v1 to v5 is 8 if k =3.
This is called the "k shortest path (routing) problem". Floyd-Warshall does not seem particularly expandable for this problem, but Dijkstra's algorithm (or if you need to handle negative weights Bellman-Ford) can be extended simply by continuing the search until k paths have been found, as they explore paths in a shortest-first way and thus the first k paths found will be the shortest.
This problem also has its own Wikipedia page with more information, which at the time of writing this also gives pseudocode for the modified Dijkstra approach: https://en.wikipedia.org/wiki/K_shortest_path_routing.