A simplex algorithm for a class of leontief flow problems

A matrix N is Leontief if it has exactly one positive entry per column and there exists a nonnegative x such that Nx > 0. A Leontief flow problem is a linear-programming problem of the form min{cTx ¦nx = b; x ≥ 0}, where N is a certain type of Leontief matrix. It is shown that for b > 0 this problem can be solved in O (n2U log npU) pivots by the simplex method using Dantzigʹs rule for choosing the entering variable, where n is the number of variables, p is the largest entry of N in absolute value, and U is a valid upper bound on any extreme-point solution. Classes of problems where this is a strongly polynomial bound are identified.