Finding the shortest path in a graph without any negative prefixes

Question

Find the shortest path from source to destination in a directed graph with positive and negative edges, such that at no point in the path the sum of edges coming

Answer 1

Admittedly this isn't a constructive answer, however it's too long to post in a comment...

It seems to me that this problem contains the binary as well as discrete knapsack problems, so its worst-case-running-time is at best pseudo-polynomial. Consider a graph that is connected and weighted as follows:

Then the equivalent binary knapsack problem is trying to choose weights from the set {a₀, ..., a_n} that maximizes Σ a_i where Σ a_i < X.

As a side note, if we introduce weighted loops it's easy to construct the unbounded knapsack problem instead.

Therefore, any practical algorithm you might choose has a running time that depends on what you consider the "average" case. Is there a restriction to the problem that I've either not considered or not had at my disposal? You seem rather sure it's an O(n³) problem. (Although what's n in this case?)