An Auction/sequential Shortest Path Algorithm for the Minimum Cost Network Flow Problem1

We propose a new algorithm for the solution of the linear minimum cost network flow problem, based on a sequential shortest path augmentation approach. Each shortest path is constructed by means of the recently proposed auction/shortest path algorithm. This approach allows useful information to be passed from one shortest path construction to the next. However, the naive implementation of this approach where the length of each arc is equal to its reduced cost fails because of the presence of zero cost cycles. We overcome this difficulty by using as arc lengths ǫ-perturbations of reduced costs and by using ǫ-complementary slackness conditions in place of the usual complementary slackness conditions. We present several variants of the main algorithm, including one that has proved very efficient for the max-flow problem. We also discuss the possibility of combining the algorithm with the relaxation method and we provide encouraging computational results. 1 Research supported by NSF under Grant CCR-9103804 and Grant 9300494-DMI. 2 Department of Electrical Engineering and Computer Science, M.I.T., Rm. 35-210, Cambridge, Mass., 02139. Email: [email protected] 1