Consider an oriented tree where each node is colored either black, white, or both. In addition, each (oriented) edge has a given weight.
I am trying to see whether there exists a pair $(u, v)$ of nodes such that both of them are black (also means that both of them can be B&W or one black and one B&W) AND
• there exists an oriented path from $u$ to $v$ such that the sum of weights on this path is equal to a given integer K.
• the sum of weights on the undirected path from $u$ to $v$ is equal to K (we are ignoring the orientation of the vertices here).
I don’t care about the vertices, I need just a YES/NO answer.