Computing optimal scalings by parametric network algorithms

A symmetric scaling of a square matrix A #0 is a matrix of the form XAX-' where X is a nonnegative, nonsingular, diagonal matrix having the same dimension of A. An asymmetric scaling of a rectangular matrix B # 0 is a matrix of the form XBY-' where X and Y"are nonnegative, nonsingular, diagonal matrices having appropriate dimensions. We consider two objectives in selecting a symmetric scaling of a given matrix. The first is to select a scaling A' of a given matrix A such that the maximal absolute value of the elements of A' is lesser or equal that of any other corresponding scaling of A. The second is to select a scaling B' of a given matrix B such that the maximal absolute value of ratios of nonzero elements of B' is lesser or equal that of any other corresponding scaling of B. We also consider the problem of finding an optimal asymmetric scaling under the maximal ratio criterion (the maximal element criterion is, of course, trivial in this case). We show that these problems can be converted to parametric network problems which can be solved by corresponding algorithms.