We introduce networks with additive losses and gains on the arcs. If a positive flow of x units enter an arc a, then x + g(a) units exit. Arcs may increase or consume flow, i.e., they are gainy or lossy. Such networks have various applications, e.g., in financial analysis, transportation, and data communication. Problems in such networks are generally intractable. In particular, the shortest path problem is NP-hard. However, there is a pseudo-polynomial time algorithm for the problem with nonnegative costs and gains. The maximum flow problem is strongly NP-hard, even in unit-gain networks and in networks with integral capacities and loss at most three, and is hard to approximate. However, it is solvable in polynomial time in unit-loss networks using the Edmonds-Karp algorithm. Our NP-hard results contrast efficient polynomial time solutions of path and flow problems in standard and in so-called generalized networks with multiplicative losses and gains.
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