Thomas L. Magnanti

A comprehensive introduction to network flows that brings together the classic and the contemporary aspects of the field, and provides an integrative view of theory, algorithms, and applications. presents in-depth, self-contained treatments of shortest path, maximum flow, and minimum cost flow problems, including descriptions of polynomial-time algorithms for these core models. emphasizes powerful algorithmic strategies and analysis tools such as data scaling, geometric improvement arguments, and potential function arguments. provides an easy-to-understand descriptions of several important data structures, including d-heaps, Fibonacci heaps, and dynamic trees. devotes a special chapter to conducting empirical testing of algorithms. features over 150 applications of network flows to a variety of engineering, management, and scientific domains. contains extensive reference notes and illustrations.

Numerous transportation applications as diverse as capital investment decision-making, vehicle fleet planning, and traffic light signal setting all involve some form of (discrete choice) network design. In this paper, we review some of the uses and limitations of integer programming-based approaches to network design, and describe several discrete and continuous choice models and algorithms. Our objectives are threefold—to provide a unifying view for synthesizing many network design models, to propose a unifying framework for deriving many network design algorithms, and to summarize computational experience in solving design problems. We also show that many of the most celebrated combinatorial problems that arise in transportation planning are specializations and variations of a generic design model. Consequently, the network design concepts described in this paper have great potential application in a wide range of problem settings.

This paper proposes methodology for improving the performance of Benders decomposition when applied to mixed integer programs. It introduces a new technique for accelerating the convergence of the algorithm and theory for distinguishing “good” model formulations of a problem that has distinct but equivalent mixed integer programming representations. The acceleration technique is based upon selecting judiciously from the alternate optima of the Benders subproblem to generate strong or pareto-optimal cuts. This methodology also applies to a much broader class of optimization algorithms that includes Dantzig-Wolfe decomposition for linear and nonlinear programs and related “cutting plane” type algorithms that arise in resource directive and price decomposition. When specialized to network location problems, this cut generation technique leads to very efficient algorithms that exploit the underlying structure of these models. In discussing the “proper” formulation of mixed integer programs, we suggest criteria for comparing various mixed integer formulations of a problem and for choosing formulations that can provide stronger cuts for Benders decomposition. From this discussion intimate connections between the previously disparate viewpoints of strong Benders cuts and tight linear programming relaxations of integer programs emerge.