A STRONGLY POLYNOMIAL TIME ALGORITHM FOR AN LP PROBLEM WITH A PRE-LEONTIEF COEFFICIENT MATRIX

Abstract

In 1991, Adler and Cosares proposed a strongly polynomial time algorithm for an LP problem with a pre-Leontief coefficient matrix and pointed out that the algorithm can be efficiently applied to a generalized transshipment problem. In their generalized transshipment problem, a given demand is satisfied at each vertex except for a distinguished one while we impose the demand condition on all the vertices. Their approach is as follows: By using Veinott’s matrix partition theorem, they partitioned the coefficient matrix into four submatrices including a Leontief submatrix, and these partitioned matrices were utilized in their algorithm. We suggest that the theorem needs more refinement. In order to clarify the suggestion, we refined the theorem to a new one by incorporating trivialities/nontrivialities of the rows and columns of a matrix whose notions were introduced by Veinott. With the help of the refined theorem, we have developed a new strongly polynomial time flow-based algorithm for a broader class of problems including their problem. In the paper by Adler and Cosares, we can not see any algorithm for finding how to divide the columns of the coefficient matrix into two sets when we partition the matrix. Given a coefficient matrix partitioned, our comlexity is the same as theirs. Our main contribution is the following two: 1) The developed algorithm can also determine the feasibility of the generalized transshipment problem, and our complexity is much smaller than theirs; 2) We showed an efficient algorithm for partitioning the given coefficient matrix into such four submatrices by introducing the trivialities/nontrivialities explained above.