Efficient generation of performance bounds for a class of traffic scheduling problems

Abstract

This work seeks to develop (lower) performance bounds for a traffic scheduling problem that arises in many application contexts, ranging from industrial material handling and robotics to computer game animations and quantum computing. In a first approach, the sought bounds are obtained by applying the Lagrangian relaxation method to a MIP formulation of the considered scheduling problem that is based on a natural notion of “state” for the underlying traffic system and an analytical characterization of all the possible trajectories of this state over a predefined time horizon. But it is also shown that the corresponding “dual” problem that provides these bounds, can be transformed to a linear program (LP) with numbers of variables and constraints polynomially related to the size of the underlying traffic system and the employed time horizon in the MIP formulation. Furthermore, the derived LP formulation constitutes the LP relaxation of a second MIP formulation for the considered scheduling problem that can be obtained through an existing connection between this problem and the “integral multi-commodity flow” (IMCF) model of network optimization theory. Finally, the theoretical developments of the paper are complemented with a computational part that demonstrates the efficacy of the pursued methods in terms of the quality of the derived bounds, and their computational tractability.