Approximating Earliest Arrival Flows in Arbitrary Networks

The earliest arrival flow problem is motivated by evacuation planning. It asks for a flow over time in a network with supplies and demands that maximizes the satisfied demands at every point in time. Gale [1959] has shown the existence of such flows for networks with a single source and sink. For multiple sources and a single sink the existence follows from work by Minieka [1973] and an exact algorithm has been presented by Baumann and Skutella [FOCS ’06]. If multiple sinks are present, it is known that earliest arrival flows do not exist in general.

We address the open question of approximating earliest arrival flows in arbitrary networks with multiple sinks and present constructive approximations of time and value for them. We give tight bounds for the best possible approximation factor in most cases. In particular, we show that there is always a 2-value-approximation of earliest arrival flows and that no better approximation factor is possible in general. Furthermore, we describe an FPTAS for computing the best possible approximation factor (which might be better than 2) along with the corresponding flow for any given instance.