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Integer programming methods for special college admissions problems

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Abstract

We develop integer programming (IP) solutions for some special college admission problems arising from the Hungarian higher education admission scheme. We focus on four special features, namely the solution concept of stable score-limits, the presence of lower and common quotas, and paired applications. We note that each of the latter three special feature makes the college admissions problem NP-hard to solve. Currently, a heuristic based on the Gale–Shapley algorithm is being used in the Hungarian application. The IP methods that we propose are not only interesting theoretically, but may also serve as an alternative solution concept for this practical application, and other similar applications. We finish the paper by presenting a simulation using the 2008 data of the Hungarian higher education admission scheme.

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Notes

  1. The 2012 Nobel-Prize in Economic Sciences was awarded to Alvin Roth and Lloyd Shapley for the theory of stable allocations and the practice of market design.

  2. The same problem has also been investigated in a master’s thesis Podhradsky (2010).

  3. In a famous study Roth (1991) analysed the nature and the long term success of a dozen resident allocation schemes established in the UK in the late seventies. He found that two schemes produced stable outcomes and both of them remained in use. From the remaining six ones, that did not always produce stable matchings, four were eventually abandoned. The two programs that were not always produced stable solutions but yet remained in use were based on linear programming techniques and has been operating in the two smallest markets. Ünver (2001) studied these programs and the possible reasons for their survival in detail.

  4. Actually until 2007 there were indeed separate quotas for the number of students in these two forms, and because of that the basic structure of the problem was different, the set system of common quotas was nested, which implied that a stable solution could be found efficiently, see details in Biró et al. (2010).

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Acknowledgments

Péter Biró: Supported by the Hungarian Academy of Sciences under its Momentum Programme (LP2016-3/2016), by OTKA grant no. K108673, and also by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Iain McBride: Supported by a SICSA Prize Ph.D. Studentship.

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Authors and Affiliations

  1. Department of Operations Research and Actuarial Sciences, Corvinus University of Budapest, Fővám tér 13-15, Budapest, 1093, Hungary

    Kolos Csaba Ágoston & Péter Biró

  2. Institute of Economics, Research Centre for Economic and Regional Studies, Hungarian Academy of Sciences, Budaörsi út 45, Budapest, 1112, Hungary

    Péter Biró

  3. School of Computing Science, University of Glasgow Sir Alwyn Williams Building, Glasgow, G12 8QQ, UK

    Iain McBride

Authors
  1. Kolos Csaba Ágoston
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  2. Péter Biró
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  3. Iain McBride
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Correspondence to Péter Biró.

Additional information

A preliminary version (Biró and McBride 2014) has been presented at COCOA 2014. This version has been significantly extended with new theoretical results and also with a completely new section on computer simulations.

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Ágoston, K.C., Biró, P. & McBride, I. Integer programming methods for special college admissions problems. J Comb Optim 32, 1371–1399 (2016). https://doi.org/10.1007/s10878-016-0085-x

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