Network Flow Algorithms

Abstract

Abstract : Network flow problems are central problems in operations research, computer science, and engineering and they arise in many real world applications. Starting with early work in linear programming and spurred by the classic book of Ford and Fulkerson, the study of such problems has led to continuing improvements in the efficiency of network flow algorithms. In spite of the long history of this study, many substantial results have been obtained within the last several years. In this survey we examine some of these recent developments and the ideas behind them. We discuss the classical network flow problems, the maximum flow problem and the minimum-cost circulation problem, and a less standard problem, the generalized flow problem, sometimes called the problem of flows with losses and gains. The survey contains six chapters in addition to this introduction. Chapter 1 develops the terminology needed to discuss network flow problems. Chapter 2 discusses the maximum flow problem, and Chapters 3, 4, and 5 discuss different aspects of the minimum-cost circulation problem, and Chapter 6 discusses the generalized flow problem. In the remainder of this introduction, we mention some of the history of network flow research, comment on some of the results to be presented in detail in later sections, and mention some results not covered in this survey. We are interested in algorithms whose running time is small as a function of the size of the network and the numbers involved (e.g. capacities, costs, or gains).