Certificates of optimality: the third way to biproportional apportionment

The biproportional apportionment problem (BAP) must be faced in many proportional electoral systems where seats must be allocated to parties within regions. BAP is a non trivial optimization problem, and only sophisticated algorithms are currently available for solving it. The issue is: are they “writable” as an actual law? Citizens rightly demand simple, easy to understand, voting systems. The alternative, though, seems to have simple, but unsound electoral laws. We propose the following way out of this dilemma: leave to a mathematically sophisticated algorithm the task of producing an optimal apportionment, but attach to it a “certificate of optimality”, that is, describe a simple procedure whereby anybody can check, through some elementary operations, that the seat allocation output by the algorithm is indeed an optimal apportionment. We discuss one such certificate, based on the Max Flow-Min Cut Theorem, relative to a parametric max flow method of ours for BAP.