Does a weighted graph have a path with weight zero? Given a weighted digraph $G=(V,E)$, where each edge is associated with a weight (could be positive, negative, or zero). We define the weight of a path to be the sum of the weights along this path. 

The question: given two vertices $s,t$, decide whether there is a (not necessarily simple) path from $s$ to $t$ such that the weight of the path is 0. 

It is not very difficult to see that this problem is in NP. I am wondering whether there is a polynomial time algorithm, or it is NP-hard.
You can reduce Subset Sum to your problem.

Rough idea: Connect s,t with chain of double edges corresponding to weights and zero. A path from s to t picks a subset.
These double edges can be expanded with dummy nodes to make a normal graph. To ensure non-empty subset make n chains to t with initial single edge corresponding to weight.