A Linear Programming Formulation of Flows over Time with Piecewise Constant Capacity and Transit Times

We present an algorithm to solve a deterministic form of a routing problem in delay tolerant networking, in which contact possibilities are known in advance. The algorithm starts with a finite set of contacts with time-varying capacities and transit delays. The output is an optimal schedule assigning messages to edges and times, that respects message priority and minimizes the overall delivery delay. The algorithm consists of two main ingredients: a discretization step in which the raw data provided by the contacts is used to obtain appropriate subdivisions of the relevant time intervals, and a linear program, a dynamic version of the classical multicommodity flow problem, in which transit times are piecewise constant, and where both edges and nodes are capacitated (in the case of edges, with piecewise constant capacities). In fact, we present two equivalent LP formulations, of which one is smaller and runs faster in CPLEX, a general purpose linear solver.