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
This paper deals with the following graph partitioning problem: given a graph with n nodes, p of which are prescribed to be centers (the remaining nodes are called units), the goal is to find a partition of the set of nodes into connected components containing only one center each, so as to minimize the total assignment cost of units to centers. This problem is known to be NP-hard in general graphs, and it is shown here to remain such even if the assignment cost is monotone and the graph is bipartite. Therefore, we specialize on tree graphs to derive polynomial time algorithms. For this class of graphs we provide several reformulations of the problem as integer linear programs. Moreover, we develop a dynamic programming algorithm, whose recursion is based on solving sequences of minimum weight closure problems, that solves the problem on trees in O(n 2 p).
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