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