On Solving Quickest Time Problems in Time-Dependent, Dynamic Networks

In this paper, a pseudopolynomial time algorithm is presented for solving the integral time-dependent quickest flow problem (TDQFP) and its multiple source and sink counterparts: the time-dependent evacuation and quickest transshipment problems. A more widely known, though less general version, is the quickest flow problem (QFP). The QFP has historically been defined on a dynamic network, where time is divided into discrete units, flow moves through the network over time, travel times determine how long each unit of flow spends traversing an arc, and capacities restrict the rate of flow on an arc. The goal of the QFP is to determine the paths along which to send a given supply from a single source to a single sink such that the last unit of flow arrives at the sink in the minimum time. The main contribution of this paper is the time-dependent quickest flow (TDQFP) algorithm which solves the TDQFP, i.e. it solves the integral QFP, as defined above, on a time-dependent dynamic network, where the arc travel times, arc and node capacities, and supply at the source vary with time. Furthermore, this algorithm solves the time-dependent minimum time dynamic flow problem, whose objective is to determine the paths that lead to the minimum total time spent completing all shipments from source to sink. An optimal solution to the latter problem is guaranteed to be optimal for the TDQFP. By adding a small number of nodes and arcs to the existing network, we show how the algorithm can be used to solve both the time-dependent evacuation and the time-dependent quickest transshipment problems.