The Generalized Shortest Path Problem

We consider generalized shortest path problems with cost and weight functions for the arcs of a graph. According to their weight arcs may generate or consume ow. Weights may be additive or multiplicative on a path. More precisely, if x units enter an arc a = (u; v) then x + w(a) resp. x w(a) leave it. The cost is charged per unit and summed over the edges of a path. If the weights multiply, then the generalized shortest path problem can be solved eeciently by the reverse Dijkstra algorithm. By a generalization we can capture consuming ows, negative costs and cost decreasing cycles. If the weights sum up, then there is a dynamic programming approach in pseudo-polynomial time. If the cost and the weight are bounded simultaneously, then we have the constraint shortest path problem, which is NP-complete. The generalized shortest path problem is a special generalized min-cost network ow problem. These arise in several application contexts and have been studied intensively in the literature, see e.g. 1]. Since the integer generalized max-ow problem is NP-hard our result gives evidence that there are no ee-cient ow augmenting path algorithms for generalized network ow problems.