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Abstract

For the earliest arrival flow problem one is given a network $G=(V, A)$ with capacities $u(a)$ and transit times $\tau(a)$ on its arcs $a \in A$, together with a source and a sink vertex $s, t \in V$. The objective is to send flow from $s$ to $t$ that moves through the network over time, such that for each point in time $\theta \in [0,T)$ the maximum possible amount of flow reaches $t$. If, for each $\theta \in [0,T)$ this flow is a maximum flow for time horizon $\theta$, then it is called \emph{earliest arrival flow}. In practical applications a higher congestion of an arc in the network often implies a considerable increase in transit time. Therefore, in this paper we study the earliest arrival problem for the case that the transit time of each arc in the network at each time $\theta$ depends on the flow on this particular arc at that time $\theta$. For constant transit times it has been shown by Gale that earliest arrival flows exist for any network. We give examples, showing that this is no longer true for flow-dependent transit times. For that reason we define an optimization version of this problem where the objective is to find flows that are almost earliest arrival flows. In particular, we are interested in flows that, for each $\theta \in [0,T)$, need only $\alpha$-times longer to send the maximum flow to the sink. We give both constant lower and upper bounds on $\alpha$; furthermore, we present a constant factor approximation algorithm for this problem. Finally, we give some computational results to show the practicability of the designed approximation algorithm.