Time-Expanded Graphs for Flow-Dependent Transit Times

Abstract

Motivated by applications in road traffic control, we study flows in networks featuring special characteristics. In contrast to classical static flow problems, time plays a decisive role. Firstly, there are transit times on the arcs of the network which specify the amount of time it takes for flow to travel through a particular arc; more precisely, flow values on arcs may change over time. Secondly, the transit time of an arc varies with the current amount of flow using this arc. Especially the latter feature is crucial for various real-life applications of flows over time; yet, it dramatically increases the degree of difficulty of the resulting optimization problems.

Most problems dealing with flows over time and constant transit times can be translated to static flow problems in time-expanded networks. We develop an alternative time-expanded network with flow-dependent transit times to which the whole algorithmic toolbox developed for static flows can be applied. Although this approach does not entirely capture the behavior of flows over time with flow-dependent transit times, we present approximation results which provide evidence of its surprising quality.

Extended abstract; information on the full version of the paper can be obtained via the authors’ WWW-pages. This work was supported in part by the EU Thematics Network APPOL I+II, Approximation and Online Algorithms, IST-1999-14084 and IST-2001-30012, by the European graduate program ‘Combinatorics, Geometry, and Computation’, Deutsche Forschungsgemeinschaft, grant GRK 588/1, and by the Bundesministerium für Bildung und Forschung (bmb+f), grant no. 03-MOM4B1.