Parametric maximum flow methods for minimax approximation of target quotas in biproportional apportionment

In this paper 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 $\ellnorm_\infty$ 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.