An Algorithm for the Detection and Construction of Monge Sequences

We give an efficient algorithm which determines whether a condition due to Hoffman (1963) is satisfied by the cost matrix of a transportation problem. In case the condition is satisfied, our algorithm generates a permutation of the matrix entries *Supported in part by Allon Fellowship and by a Bat Sheva de Rothschild grant. ‘Supported in part by the National Science Foundation under grant ECS-G-01988 and the Office of Naval Research under grant NGOO 1488-KXI377. *Supported in part by Allon Fellowship. LINEAR ALGEBRA AND ITS APPLICATIONS 114/115:669-680 (1989) B Elsevier Science Publishing Co., Inc., 1989 669 655 Avenue of the Americas, New York, NY 10010 00243795/89/$3.50 670 NOGA ALON ET AL. (called a Monge sequence), which allows for the solution of any problem with that cost matrix in linear time, by way of a “greedy” algorithm. This is the first polynomial algorithm for this problem. The running time of our algorithm is better than that of the best known algorithms for the transportation problem, and thus it can be used as a preliminary step in solving such problems without an increase in the overall complexity.