Distance-Directed Augmenting Path Algorithms for Maximum Flow and Parametric Maximum Flow Problems Ravindra K. Ahuja* Department of Industrial and Management Engineering, Indian Institute of Technology, Kanpur-208 016, India James B. Orlin Sloan School of Management, MIT, Cambridge, Massachusetts 02139 Until recently, fast algorithms for the maximum flow problem have typically pro- ceeded by constructing layered networks and establishing blocking flows in these networks. However, in recent years, new distance-directed algorithms have been suggested that do not construct layered networks but instead maintain a distance label with each node. The distance label of a node is a lower bound on the length of the shortest augmenting path from the node to the sink. In this article we develop two distance-directed augmenting path algorithms for the maximum flow problem. Both the algorithms run in O(n 2 m) time on networks with n nodes and m arcs. We also point out the relationship between the distance labels and layered networks. Using a scaling technique, we improve the complexity of our distance-directed algorithms to O(nm log U), where U denotes the largest arc capacity. We also consider applications of these algorithms to unit capacity maximum flow problems and a class of parametric maximum flow problems. 1. INTRODUCTION In this article we suggest several new algorithms for the maximum flow and parametric maximum flow problems. The maximum flow problem is one of the most fundamental network flow problems and arises in a variety of situations. A recent paper of Ahuja, Orlin, and Reddy [4] describes about 25 applications of the maximum flow problem. Besides these applications, there are other more complex optimization problems whose algorithms use the maximum flow algo- rithm as a subroutine. Some of these problems are the time-cost tradeoff problem in CPM networks (Fulkerson [13], Kelley [24]), the parametric network feasi- bility problem (Mineka [26]), the network design problem (Hu [21]), and the minimax transportation problem (Ahuja [1]). Moreover, the maximum flow *This work was performed when the author was visiting the Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. Naval Research Logistics, Vol. 38, pp. 413-430 (1991) Copyright © 1991 by John Wiley & Sons, Inc. CCC 0028-1441/91/030413-18$04.00 i ) 0 :~ ~ ~~~~~_ _______ -- I·I·LIIL1g) - _ qplP- CI- L---------------l"--CI---·-CLI-nLP--CI -- Ea- ·· -r------·----- t i i i 7 i i i I '' ' L '' .pi i: . 1. . I I. .. ." . I i -I'. --A i i I i f i II .i I