graph is oriented
each edge has an assigned value representing the cost of using this edge. The value can be positive or negative
input: graph, starting node and initial value (I don't have a goal node)
both, nodes and edges can be used repeatedly
goal: change the initial value to zero by passing through the graph. The answer should be if it is possible to reach zero (exactly zero, without reaching a negative actual value in the process)
I don't need the final path as the result, just the information if it is possible is enough. I would be most interested in a name of algorithm that is designed for this problem.
It is clearly NP-Hard (subset-sum can be reduced to it by using an appropriate complete graph). Breadth-first search seems like a natural approach, though to get a decision-procedure out of it you would need to find an upper-bound on the length of the path.