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In contrast to traditional flow networks, in additive flow networks, to everyedge e is assigned a gain factor g(e) which represents the loss or gain of theflow while using edge e. Hence, if a flow f(e) enters the edge e and f(e) isless than the designated capacity of e, then f(e) + g(e) = 0 units of flowreach the end point of e, provided e is used, i.e., provided f(e) != 0. In thisreport we study the maximum flow problem in additive flow networks, which weprove to be NP-hard even when the underlying graphs of additive flow networksare planar. We also investigate the shortest path problem, when to every edge eis assigned a cost value for every unit flow entering edge e, which we show tobe NP-hard in the strong sense even when the additive flow networks are planar.

Shortest path and maximum flow problems in planar flow networks with additive gains and losses