Judge whether exist a simple path visiting certain node?

Question

Given a graph <V, E>, starting node st, end node ed and certain node set M that must be visited. My question is to find the simple path that visiting all the M. Also I want to know:

1 Is the path exist?

2 If exists, how to find it as fast as possible?

When M=V you get the Hamiltonian path problem, so it is not solvable "fast" in the general case. — n. m. could be an AI, Mar 27 '16 at 14:18
Thanks @n.m. But can we get approximate or randomized algorithm to solve it? — mickeyandkaka, Mar 27 '16 at 14:26
How do you approximate a path that visits all the vertices? Should an approximate path visit most of the vertices or what? I have no idea either way, just curious. — n. m. could be an AI, Mar 27 '16 at 14:30
May the path contain only vertices of `M`? May a vertex be visited twice? — lex82, Mar 27 '16 at 14:33
I means use genetic algorithm or other algorithm (which may not get the correct answer) like it to solve it. @lex82 simple path means every node must be visited at most once. Path can contain other nodes. — mickeyandkaka, Mar 27 '16 at 14:35

score 1 · Answer 1 · answered Mar 28 '16 at 10:32

Because of the set M the problem can be reduced to Hamiltonian path with is known to be NP-Complete.

The way to tell if a path exists is to find one, and to find one the fastest thing is N! in the number of nodes.

What you can do is find fast if a path doesn't exist (disconnected components or critical nodes that need to be passed more than once in order to complete the path) and have some smart heuristics to stop early in the recursion (when it becomes obvious that no path can be completed that way) or pick better node order.

Judge whether exist a simple path visiting certain node?

1 Answers1