Heuristic for using A* to find the path with the highest gain

Question

Suppose that I want to change the logic in A*, trying to find the most useful path (i.e., the one with the highest gain) instead of finding the shortest path (i.e., the one with

Answer 1

Your problem is the longest path problem, which is strongly NP-Hard. This means that, not only is there (almost certainly) no fast exact algorithm, but there is also (almost certainly) no good approximate algorithm.

You will unfortunately either have to brute-force it, or resort to various global optimization techniques, like annealing, genetic programming etc.

---

Negating the sign of the edge-weights, as @Charles suggests, will not work, as A* cannot handle negative edge-weights. And other algorithms which can handle negative edge-weights still cannot handle negative cycles.