Vector and matrix apportionment problems and separable convex integer optimization

Algorithms for the proportional rounding of a nonnegative vector, and for the biproportional rounding of a nonnegative matrix are discussed. Here we view vector and matrix rounding as special instances of a generic optimization problem that employs an additive version of the objective function of Gaffke and Pukelsheim (2007). The generic problem turns out to be a separable convex integer optimization problem, in which the linear equality constraints are given by a totally unimodular coefficient matrix. So, despite the integer restrictions of the variables, Fenchel duality applies. Our chief goal is to study the implied algorithmic consequences. We establish a general algorithm based on the primal optimization problem. Furthermore we show that the biproportional algorithm of Balinski and Demange (1989), when suitably generalized, derives from the dual optimization problem. Finally we comment on the shortcomings of the alternating scaling algorithm, a discrete variant of the well-known Iterative Proportional Fitting procedure

Network flow methods for electoral systems

Researchers in the area of electoral systems have recently turned their attention to network flow techniques with the aim to resolve certain practically relevant problems arising in this area. The aim of this paper is to review some of this work, showing the applicability of these techniques even to problems of a very different nature. Major emphasis will be placed on biproportional apportionment, a problem that frequently arises in proportional electoral systems, but which in some countries is still ill-solved, or not dealt with rigorously, notwithstanding the availability of several sound solution procedures and their concrete application in some real-life elections. Besides biproportional apportionment, we shall discuss applications of network flows to problems such as vote transitions and political districting. Finally, we address the so-called give-up problem, which arises in the current elections for the Italian Parliament. It is related to the possible assignment of seats to multiple winners of a given party. Based on the results and techniques presented in this article, it is fair to state that network flow models and algorithms are indeed very flexible and effective tools for the analysis and the design of contemporary electoral systems. (C) 2011 Wiley Periodicals, Inc. NETWORKS, Vol. 59(1), 73-88 201