Reduced costs obtained from the optimal simplex tableau are not necessarily correct under degeneracy in network flow problems with side constraints. In one application, simplex reduced costs were shown not to be good when applied to actual operations, while reoptimization to find all true reduced costs was too time-consuming. An efficient algorithm to obtain approximate reduced costs was developed in the research to offset these disadvantages. The algorithm combines the uses of Lagrangian relaxation, a minimum-cost network flow algorithm, and a shortest path algorithm. In theory, the approximate reduced costs prove to be a better lower bound than are simplex reduced costs. Numerical experiments have been performed to show its potential usefulness in practice. (C) 1996 John Wiley & Sons, Inc.