Algorithms for proportional matrices in reals and integers
Mathematical Programming: Series A and B
An axiomatic approach to proportionality between matrices
Mathematics of Operations Research
Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
Polynomial Methods for Separable Convex Optimization in Unimodular Linear Spaces with Applications
SIAM Journal on Computing
Lindo an Optimization Modeling System/Book and Macintosh Disk
Lindo an Optimization Modeling System/Book and Macintosh Disk
Matrix scaling by network flow
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
On the complexity of general matrix scaling and entropy minimization via the RAS algorithm
Mathematical Programming: Series A and B
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In this article, we study the biproportional apportionment problem, which deals with the assignment of seats to parties within regions. We consider the minimization of both the maximum absolute error and the maximum relative error of the apportioned seats with respect to target quotas. We show that this can be done polynomially through a reduction to a parametric maximum flow problem. Moreover, the maximum absolute error can be minimized in strongly polynomial time. More generally, our method can be used for computing $\ell_{\infty}$ **image** projections onto a flow polytope. We also address the issue of uniqueness of the solution, proposing a method based on finding unordered lexicographic minima. Our procedure is compared to other well-known ones available in the literature. Finally we apply our procedures to the data of the 2008 Italian political elections, for which the procedure stated by the law produced an inconsistent assignment of seats. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012 © 2012 Wiley Periodicals, Inc.