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Given an n × m nonnegative matrix A = (a_{ij}) and positive integral vectors r in R^n and c in R^m having a common one-norm h, the (r, c)-scaling problem is to obtain positive diagonal matrices X and Y, if they exist, such that XAY has row and column sums equal to r and c, respectively. The entropy minimization problem corresponding to A is to find an n × m matrix z = (z_{ij}) having the same zero pattern as A, the sum of whose entries is a given number h, its row and column sums are within given integral vectors of lower and upper bounds, and such that the entropy function consisting of the sum of the terms z_{ij} ln(z_{ij}/a_{ij}) is minimized. When the lower and upper bounds coincide, matrix scaling and entropy minimization are closely related. In this paper we present several complexity bounds for the \epsilon-approximate (r, c)-scaling problem, polynomial in n, m, h, 1/\epsilon, and ln V , where V and v are the largest and the smallest positive entries of A, respectively. These bounds, although not polynomial in ln(1/\epsilon), not only provide alternative complexities for the polynomial time algorithms, but could result in better overall complexities. In particular, our theoretical analysis supports the practicality of the well-known RAS algorithm

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The political districting problem has been studied since the 60's and many different models and techniques have been proposed with the aim of preventing districts' manipulation which may favor some specific political party (gerrymandering). A variety of political districting models and procedures was provided in the Operations Research literature, based on single- or multiple-objective optimization. Starting from the forerunning papers published in the 60's, this article reviews some selected optimization models and algorithms for political districting which gave rise to the main lines of research on this topic in the Operartions Research literature of the last five decades