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Vector and matrix apportionment problems and separable convex integer optimization

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  • N. Gaffke
  • F. Pukelsheim

Abstract

The problems of (bi-)proportional rounding of a nonnegative vector or matrix, resp., are written as particular separable convex integer minimization problems. Allowing any convex (separable) objective function we use the notions of vector and matrix apportionment problems. As a broader class of problems we consider separable convex integer minimization under linear equality restrictions Ax = b with any totally unimodular coefficient matrix A. By the total unimodularity Fenchel duality applies, despite the integer restrictions of the variables. The biproportional algorithm of Balinski and Demange (Math Program 45:193–210, 1989) is generalized and derives from the dual optimization problem. Also, a primal augmentation algorithm is stated. Finally, for the smaller class of matrix apportionment problems we discuss the alternating scaling algorithm, which is a discrete variant of the well-known Iterative Proportional Fitting procedure. Copyright Springer-Verlag 2008

Suggested Citation

  • N. Gaffke & F. Pukelsheim, 2008. "Vector and matrix apportionment problems and separable convex integer optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(1), pages 133-159, February.
    • Handle: RePEc:spr:mathme:v:67:y:2008:i:1:p:133-159
    DOI: 10.1007/s00186-007-0184-7
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    References listed on IDEAS

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    1. H. J. J. te Riele, 1978. "The proportional representation problem in the Second Chamber: an approach via minimal distances," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 32(4), pages 163-179, December.
    2. Michel L. Balinski & Gabrielle Demange, 1989. "Algorithm for Proportional Matrices in Reals and Integers," Post-Print halshs-00585327, HAL.
    3. Michel Balinski, 2006. "Apportionment: Uni- and Bi-Dimensional," Studies in Choice and Welfare, in: Bruno Simeone & Friedrich Pukelsheim (ed.), Mathematics and Democracy, pages 43-53, Springer.
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    Citations

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    Cited by:

    1. Oelbermann, Kai-Friederike, 2016. "Alternate Scaling algorithm for biproportional divisor methods," Mathematical Social Sciences, Elsevier, vol. 80(C), pages 25-32.
    2. Niemeyer, Horst F. & Niemeyer, Alice C., 2008. "Apportionment methods," Mathematical Social Sciences, Elsevier, vol. 56(2), pages 240-253, September.
    3. Kerem Akartunalı & Philip A. Knight, 2017. "Network models and biproportional rounding for fair seat allocations in the UK elections," Annals of Operations Research, Springer, vol. 253(1), pages 1-19, June.
    4. Paolo Serafini, 2015. "Certificates of optimality for minimum norm biproportional apportionments," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 44(1), pages 1-12, January.
    5. Friedrich Pukelsheim, 2014. "Biproportional scaling of matrices and the iterative proportional fitting procedure," Annals of Operations Research, Springer, vol. 215(1), pages 269-283, April.
    6. Gaffke, Norbert & Pukelsheim, Friedrich, 2008. "Divisor methods for proportional representation systems: An optimization approach to vector and matrix apportionment problems," Mathematical Social Sciences, Elsevier, vol. 56(2), pages 166-184, September.

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