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Given an n × m nonnegative matrix A =(a ij ) and positive integral vectorsr ε ℜ n and c εÂm having a common one-norm h, the(r,c)-scaling problem is to obtain positive diagonalmatrices X and Y, if they exist, such that XAYhas row and column sums equal to r and c,respectively. The entropy minimization problem corresponding toA is to find an n × m matrix z =(z ij ) having the same zero pattern asA, the sum of whose entries is a given number h, itsrow and column sums are within given integral vectors of lower andupper bounds, and such that the entropy function consisting of thesum of the terms z ij ln (zij /a ij ) is minimized.When the lower and upper bounds coincide, matrix scaling andentropy minimization are closely related. In this paper we presentseveral complexity bounds for the ε-approximate(r,c)-scaling problem, polynomial inn,m,h, 1/ε, and ln V/v,where V and v are the largest and the smallestpositive entries of A, respectively. These bounds, althoughnot polynomial in ln(1/1/ε), not only provide alternativecomplexities for the polynomial time algorithms, but could resultin better overall complexities. In particular, our theoreticalanalysis supports the practicality of the well-known RAS algorithm.In our analysis we obtain bounds on the norm of scaling vectorswhich will be used in deriving not only some of the abovecomplexities, but also a complexity for square nonnegative matriceshaving positive permanent. In particular, our results extend,nontrivially, many bounds for the doubly stochastic scaling ofsquare nonnegative matrices previously given by Kalantari andKhachiyan to the case of general (r,c)-scaling.Finally, we study a more general entropy minimization problem whererow and column sums are constrained to lie in prescribed intervals,and the sum of all entries is also prescribed. Balinski and Demangedescribed an RAS type algorithm for its continuous version, but didnot analyze its complexity. We show that this algorithm produces an1/ε-approximate solution within complexity polynomial inn, m, h, V/v and1/1/ε.