Generalized Maximum Flows over Time

Flows over time and generalized ows are two advanced network ow models of utmost importance, as they incorporate two crucial features occurring in numerous real-life networks. Flows over time feature time as a problem dimension and allow to realistically model the fact that commodities (goods, information, etc.) are routed through a network over time. Generalized ows allow for gain/loss factors on the arcs that model physical transformations of a commodity due to leakage, evaporation, breeding, theft, or interest rates. Although the latter e ects are usually time-bound, generalized ow models featuring a temporal dimension have never been studied in the literature. In this paper we introduce the problem of computing a generalized maximum ow over time in networks with both gain factors and transit times on the arcs. While generalized maximum ows and maximum ows over time can be computed e ciently, our combined problem turns out to be NP-hard and even completely non-approximable. A natural special case is given by lossy networks where the loss rate per time unit is identical on all arcs. For this case we present a (practically e cient) FPTAS that also reveals a surprising connection to so-called earliest arrival ows.