Question

I have a tree where each edge is assigned a weight (a real number that can be positive or negative). I need an algorithm to find a simple path of maximum total weight (that is, a simple path where the sum of the weights of the edges in the path is maximum). There’s no restriction on what node the path starts or ends.

I have a possible algorithm, but I am not sure it works and I am looking for a proof. Here it is:

1. Select an arbitrary node u and run DFS(u) to find the maximum weight simple path that starts at u. Let (u, v) be this path.
2. Run DFS(v) to find the maximum weight simple path that starts at v. Let this path be (v, z).

Then (v, z) is a simple path of maximum weight. This algorithm is linear in the size of the graph. Can anyone tell me if it works, and if so, give a proof?

Note: The Longest Path Problem is NP-Hard for a general graph with cycles. However, I only consider trees here.

Answer 1

If you allow negative weights then consider the following example:

a<->b : -5000
a<->c : 1
b<->d : 1
b<->e : 1

The longest path is d<->b<->e with length 2

Arbitrarily pick a to start. DFS returns c with a distance of 1. Second DFS returns a with a distance of 1. However a<->c is not the longest path.