Shortest path and maximum flow problems in planar flow networks with additive gains and losses
In contrast to traditional flow networks, in additive flow networks, to every edge e is assigned a gain factor g(e) which represents the loss or gain of the flow while using edge e. Hence, if a flow f(e) enters the edge e and f(e) is less than the designated capacity of e, then f(e) g(e) 0 units of flow reach the end point of e, provided e is used, i.e. provided f(e) 0. In this report we study the maximum flow problem in additive flow networks, which we prove to be NP hard even when the underlying graphs of additive flow networks are planar. We also investigate the shortest path problem, when to every edge e is assigned a cost value for every unit flow entering edge e, which we show to be NP hard in the strong sense even when the additive flow networks are planar.
ArXiv