How to find the longest simple path in a graph?

Question

I know that for non-directed graph this problem is NP-complete hence we should do Brute Force in order to check all possible paths. How we can do that? Please suggest a pseu

Answer 1

If your graph is a special case in which it's directed and acyclic, you could do a dynamic programming approach such as the one described here. You basically sort your graph topologically, then in the topological order, for every node V, you check all its neighbors and update their "distance" value if it's bigger than the "distance" already memorized (initialized with -infinity or something).

Otherwise, in the general case, the problem is indeed NP-complete as it reduces to the Hamiltonian cycle. One thing you could do is negate all the edges and try the Bellman-Ford Algorithm. Beware that it's not good for negative cycles, however.