Digital Object Identifier (DOI) 10.1007/s101070100248 Math. Program., Ser. A 91: 271–288 (2002) Donald Goldfarb · Zhiying Jin · Yiqing Lin A polynomial dual simplex algorithm for the generalized circulation problem ⋆ Received: June 1998 / Accepted: June 27, 2001 Published online September 17, 2001 –  Springer-Verlag 2001 Abstract. This paper presents a polynomial-time dual simplex algorithm for the generalized circulation problem. An efficient implementation of this algorithm is given that has a worst-case running time of O(m 2 (m + n log n) log B), where n is the number of nodes, m is the number of arcs and B is the largest integer used to represent the rational gain factors and integral capacities in the network. This running time is as fast as the running time of any combinatorial algorithm that has been proposed thus far for solving the generalized circulation problem. Key words. network flows – generalized circulation – generalized maximum flow – dual simplex algorithm – arc excess – excess scaling 1. Introduction The generalized circulation problem, also known as the generalized maximum flow problem, is a generalization of the ordinary maximum flow problem. In the latter prob- lem, the goal is to find a flow in a directed network that maximizes the net amount of flow into a special node, the sink, while satisfying capacity constraints on all arcs and flow conservation at all nodes other than the sink and another specified node, the source. In the generalized circulation problem, there is no distinguished source node and the flow along an arc may increase or decrease by a fixed factor during its traversal of that arc. Specifically, a gain factor γ(v, w) > 0 is associated with each arc (v, w) in the network so that if one unit of flow leaves node v along arc (v, w), then γ(v, w) units of flow enter node w. Since gain factors can be used to describe phenomena such as currency exchange rates, failure rates, transportation and transmission losses, and transformations of one good to another, the generalized circulation problem can be used to model problems in finance, reliability, transportation, communication and manufacturing. For specific applications see [1], [6] and [8]. In this paper we present a polynomial-time dual simplex algorithm for the general- ized circulation problem that is based upon one of the polynomial-time combinatorial algorithms of Goldfarb, Jin and Orlin [13]. Consequently, we shall use notation that is for the most part the same as that used in [13]. Let G = (V, E, u , γ, s) be a directed ⋆ This research was supported in part by NSF Grants DMS 94-14438, DMS 95-27124 and CDA 97-26385 and DOE Grant DE-FG02-92ER25126 D. Goldfarb: Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027 Z. Jin: GTE Laboratories, 40 Sylvan Road, Waltham, MA 02154 Y. Lin: United Technologies Research Center, 411 Silver Lane, East Hartford, CT 06108