无向非加权图中的最长路径

Question

I came across a problem where I have to find out the longest path in a given graph. I have list of edges ( eg.{AB, BC} ) which states there is an edge between vertices/nodes (A,B,C). Now i want to figure out the longest path possible (not repeating the vertex) such that it covers maximum nodes starting from any vertex/node.

What can be the best way to solve this? I have to implement this as a program.

I looked up google for Minimum Spanning Tree, Dijkstra's Alogorithms , and many more. but can't figure out what would suit best for this problem.

Any help or reading references would be much appreciated.

Answer 1

This is NP-Hard.

You can solve Hamiltonian Path problem, if you can solve this problem.

(Longest path = |V| <=> Hamiltonial Path).

So, just pick up any algorithm for Travelling Salesman Problem. Give all the edges weight one.

I heard a dynamic programming version is particularly nice.

Answer 2

Similar to finding Hamiltonian paths, this needs to be solved by backtracking.

For each vertex in the graph, initialize a path list. Each element of path will contain a vertex, and a list of vertices ( adj ) which can be reached from that vertex. That is, path[i] -> (v, adj) . We'll make it a rule that the vertices in adj[v] must not be vertices that occur earlier in path .

path[i+1] is built from path[i] by letting path[i+1].v = pop(path[i].adj) , and setting path[i+1].adj equal the adjacent vertices of path[i+1].v that are not yet in the path. If there is no element of path[i].adj to pop, we've reached a dead end. If the path is longer than the largest path found so far record it.

Now pop all the elements from path until you find one that doesn't have an empty adj list (this is called backtracking). After backtracking, either the last path element has a non-empty adj list, or the path is empty, ie you've backtracked to the beginning. If the path is not empty, extend the path as before. If the path is empty, choose a new starting vertex and begin again.

Eventually all possible paths are enumerated, so the longest path found is your answer.

You may be able to find ways to short-cut this process, possibly using cut-edges (aka bridge edges).

Answer 3

Since your question doesn't speak whether the Graph is Cyclic or Not you have two options:

Option1 : Graph is DAG

You are Lucky, you can use topological sort on the graph and get the longest path!

Option 2 : Graph is NOT DAG:

Use the Hamiltonian Algorithm as mentioned in the Comments!

Answer 4

Give all edges weight 1. The non-existent, give weight n+1. Now apply TSP.

Leave something for the OP to work out, but alas apparently this was harder than I assumed it was.