Network Flow Algorithms

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 [26], 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'trie 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. 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). As a measure of the network size, we use n to denote the number of vertices and m to denote the number of arcs. As measures of the number sizes, we use U to denote the maximum arc capacity, C to denote the maximum arc cost, and B (in the case of the generalized flow problem) to denote the maximum numerator or denominator of the arc capacities and gains. In bounds using U, C, or B, we make the assumption that the capacities and costs are integers, and that the gains in the generalized flow problem are rational numbers. We are most interested in polynomial-time algorithms. We make the following distinctions. An algorithm is pseudopolynomial if its running time has a bound that is polynomial in n,m, and the appropriate subset of U, C, and B. An algorithm is polynomial if its running time has a bound that is polynomial in n,m,