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Network flow methods for electoral systems
, 2011
"... 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 the present paper is review some of this work, showing the applicability of these techniques eve ..."
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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 the present paper is 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 illsolved, or not dealt with rigorously, notwithstanding the availability of several sound solution procedures and their concrete application in some reallife elections. Beside biproportional apportionment, we shall discuss applications of network flows to problems such as vote transitions and political districting. Finally, we address the socalled “Giveup 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 paper, 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.
On the iterative proportional fitting procedure: Structure of accumulation points and L1error analysis
, 2009
"... A new analysis of the Iterative Proportional Fitting procedure is presented. The input data consist of a nonnegative matrix, and of row and column marginals. The output sought is a biproportional fit, that is, a scaling of the input matrix by means of row and column divisors so that the scaled matr ..."
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A new analysis of the Iterative Proportional Fitting procedure is presented. The input data consist of a nonnegative matrix, and of row and column marginals. The output sought is a biproportional fit, that is, a scaling of the input matrix by means of row and column divisors so that the scaled matrix has row and column sums equal to the input marginals. The IPF procedure is an algorithm alternating between the fitting of rows and columns. The structure of its accumulation points is explored in detail. The progress of the algorithm is evaluated through an L1error function measuring the deviation of current row and column sums from target marginals. A formula is obtained, of maxflow mincut type, to calculate the minimum L1error directly from the input data. If the minimum L1error is zero, the IPF procedure converges to the unique biproportional fit. Otherwise, it eventually oscillates between various accumulation points.
DivisorBased Biproportional Apportionment in Electoral Systems: A RealLife Benchmark Study
, 2007
"... Biproportional apportionment methods provide twoway proportionality in electoral systems where the electoral region is subdivided into electoral districts. The problem is to assign integral values to the elements of a matrix that are proportional to a given input matrix, and such that a set of row ..."
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Biproportional apportionment methods provide twoway proportionality in electoral systems where the electoral region is subdivided into electoral districts. The problem is to assign integral values to the elements of a matrix that are proportional to a given input matrix, and such that a set of row and columnsum requirements are fulfilled. In a divisorbased method for biproportional apportionment the problem is solved by computing appropriate row and columndivisors, and by rounding the quotients. We present a comprehensive experimental evaluation of divisorbased biproportional apportionment in an electoral system context. Firstly, we study the practical performance of a range of algorithms by performing experiments on reallife benchmark instances (election data with multimember districts). Secondly, we evaluate the general quality of divisorbased apportionments with respect to, e.g., deviation from quota, reversal orderings and occurrences of ties.
Network Models and Biproportional Apportionment for Fair Seat Allocations in the UK Elections
, 2013
"... Systems for allocating seats in an election offer a number of socially and mathematically interesting problems. We discuss how to model the allocation process as a network flow problem, and propose a wide choice of objective functions and allocation schemes. We present biproportional apportionment ..."
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Systems for allocating seats in an election offer a number of socially and mathematically interesting problems. We discuss how to model the allocation process as a network flow problem, and propose a wide choice of objective functions and allocation schemes. We present biproportional apportionment, which is an instance of the network flow problem and is used in some European countries with multiseat constituencies. We discuss its application to single seat constituencies and the inevitable consequence that seats are allocated to candidates with little local support. However, we show that variants can be selected, such as regional apportionment, to mitigate this problem. In particular, we introduce a parameter based family of methods which can be tuned to meet the public’s demand for local and global “fairness”. Using data from the 2010 UK General Election, we study a variety of network models and biproportional apportionments, and address conditions of existence and uniqueness.
Minimum ratio cover of matrix columns by extreme rays of its induced cone
"... Abstract. Given a matrix S ∈ R m×n and a subset of columns R, we study the problem of finding a cover of R with extreme rays of the cone , where an extreme ray v covers a column k if v k > 0. In order to measure how proportional a cover is, we introduce two different minimization problems, namel ..."
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Abstract. Given a matrix S ∈ R m×n and a subset of columns R, we study the problem of finding a cover of R with extreme rays of the cone , where an extreme ray v covers a column k if v k > 0. In order to measure how proportional a cover is, we introduce two different minimization problems, namely the minimum global ratio cover (MGRC) and the minimum local ratio cover (MLRC) problems. In both cases, we apply the notion of the ratio of a vector v, which is given by . These problems are originally motivated by a biological question on metabolic networks. We show that these two problems are NPhard, even in the case in which R = 1. We introduce a mixed integer programming formulation for the MGRC problem, which is solvable in polynomial time if all columns should be covered, and introduce a branchandcut algorithm for the MLRC problem. Finally, we present computational experiments on data obtained from real metabolic networks.