Advertisement

Journal of Combinatorial Optimization

, Volume 32, Issue 4, pp 1371–1399 | Cite as

Integer programming methods for special college admissions problems

  • Kolos Csaba Ágoston
  • Péter Biró
  • Iain McBride
Article

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.

Keywords

College admissions problem Integer programming Stable score-limits Lower quotas Common quotas Paired applications Simulations 

Mathematics Subject Classification

C61 C63 C78 

Notes

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.

References

  1. Abraham D, Blum A, Sandholm T (2007) Clearing algorithms for Barter-Exchange Markets: enabling nationwide kidney exchanges. In: Proceedings of ACM-EC 2007: the eighth ACM conference on electronic commerce. ACM, New York, pp 295–304Google Scholar
  2. Abdulkadiroglu A, Ehlers L (2007) Controlled school choice. Working paperGoogle Scholar
  3. Azavedo EM, Leshno JD (2012) A supply and demand framework for two-sided matching markets. Working paperGoogle Scholar
  4. Baïou M, Balinski M (2000) The stable admissions polytope. Math Program 87(3):427–439MathSciNetCrossRefMATHGoogle Scholar
  5. Biró P (2008) Student admissions in Hungary as Gale and Shapley envisaged. Technical report, no. TR-2008-291 of the Computing Science Department of Glasgow UniversityGoogle Scholar
  6. Biró P (2012) University admission practices—Hungary. http://www.matching-in-practice.eu. Accessed 23 May 2012
  7. Biró P, Fleiner T, Irving RW, Manlove DF (2010) The college admissions problem with lower and common quotas. Theoret Comput Sci 411:3136–3153MathSciNetCrossRefMATHGoogle Scholar
  8. Biró P, Irving RW, Schlotter I (2011) Stable matching with couples—an empirical study. ACM J Exp Algorithm 16:1–2MathSciNetMATHGoogle Scholar
  9. Biró P, Kiselgof S (2015) College admissions with stable score-limits. CEJOR 23(4):727–741MathSciNetCrossRefMATHGoogle Scholar
  10. Biró P, Klijn F (2013) Matching with couples: a multidisciplinary survey. Int Game Theory Rev 15(2):1340008MathSciNetCrossRefMATHGoogle Scholar
  11. Biró P, McBride I (2014) Integer programming methods for special college admissions problems. In: Proceedings of COCOA 2014: the 8th annual international conference on combinatorial optimization and applications. LNCS, vol 8881. Springer, Berlin, pp 429–443Google Scholar
  12. Biró P, McBride I, Manlove DF (2014) The hospitals/residents problem with couples: complexity and integer programming models. In: Proceedings of SEA 2014: the 13th international symposium on experimental algorithms. LNCS, vol 8504. Springer, New York, pp 10–21Google Scholar
  13. Fleiner T (2003) On the stable \(b\)-matching polytope. Math Soc Sci 46:149–158MathSciNetCrossRefMATHGoogle Scholar
  14. Fleiner T, Jankó Zs (2014) Choice function-based two-sided markets: stability, lattice property, path independence and algorithms. Algorithms 7(1):32–59MathSciNetCrossRefGoogle Scholar
  15. Gale D, Shapley LS (1962) College admissions and the stability of marriage. Am Math Mon 69(1):9–15MathSciNetCrossRefMATHGoogle Scholar
  16. Gale D, Sotomayor MAO (1985) Some remarks on the stable matching problem. Discret Appl Math 11(3):223–232MathSciNetCrossRefMATHGoogle Scholar
  17. Irving RW (2008) Stable matching problems with exchange restrictions. J Comb Optim 16:344–360MathSciNetCrossRefMATHGoogle Scholar
  18. Irving RW, Manlove DF (2008) Approximation algorithms for hard variants of the stable marriage and hospitals/residents problems. J Comb Optim 16:279–292MathSciNetCrossRefMATHGoogle Scholar
  19. Kamada Y, Kojima F (2012) Stability and strategy-proofness for matching with constraints: a problem in the Japanese medical match and its solution. Am Econ Rev 102(3):366–370CrossRefGoogle Scholar
  20. Kwanashie A, Manlove DF (2014) An integer programming approach to the hospitals / residents problem with ties. In: Proceedings of OR 2013: the international conference on operations research. Springer, Berlin, pp 263–269Google Scholar
  21. McDermid EJ, Manlove DF (2012) Keeping partners together: algorithmic results for the hospitals/residents problem with couples. J Comb Optim 19:279–303MathSciNetCrossRefMATHGoogle Scholar
  22. Manlove DF, O’Malley G (2012) Paired and altruistic kidney donation in the UK: algorithms and experimentation. In: Proceedings of SEA 2012: the 11th international symposium on experimental algorithms. LNCS, vol 7276. Springer, Berlin, pp 271–282Google Scholar
  23. Podhradsky A (2010) Stable marriage problem algorithms. Master’s thesis, Faculty of Informatics, Masaryk UniversityGoogle Scholar
  24. Ronn E (1990) NP-complete stable matching problems. J Algorithms 11:285–304MathSciNetCrossRefMATHGoogle Scholar
  25. Roth AE (1984) The evolution of the labor market for medical interns and residents: a case study in game theory. J Polit Econ 92(6):991–1016CrossRefGoogle Scholar
  26. Roth AE (1991) A natural experiment in the organization of entry-level labor markets: regional markets for new physicians and surgeons in the United Kingdom. Am Econ Rev 81:415–440Google Scholar
  27. Roth AE (1986) On the allocation of residents to rural hospitals: a general property of two-sided matching markets. Econometrica 54(2):425–427MathSciNetCrossRefGoogle Scholar
  28. Roth AE, Rothblum UG, Vande Vate JH (1993) Stable matchings, optimal assignments, and linear programming. Math Oper Res 18(4):803–828MathSciNetCrossRefMATHGoogle Scholar
  29. Roth AE, Sönmez T, Ünver MU (2007) Efficient kidney exchange: coincidence of wants in markets with compatibility-based preferences. Am Econ Rev 97(3):828–851CrossRefGoogle Scholar
  30. Rothblum UG (1992) Characterization of stable matchings as extreme points of a polytope. Math Program 54(1, Ser. A):57–67MathSciNetCrossRefMATHGoogle Scholar
  31. Sethuraman J, Teo C-P, Qian L (2006) Many-to-one stable matching: geometry and fairness. Math Oper Res 31(3):581–596MathSciNetCrossRefMATHGoogle Scholar
  32. Ünver MU (2001) Backward unraveling over time: the evolution of strategic behavior in the entry-level British medical labor markets. J Econ Dyn Control 25:1039–1080CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Operations Research and Actuarial SciencesCorvinus University of BudapestBudapestHungary
  2. 2.Institute of Economics, Research Centre for Economic and Regional StudiesHungarian Academy of SciencesBudapestHungary
  3. 3.School of Computing ScienceUniversity of Glasgow Sir Alwyn Williams BuildingGlasgowUK

Personalised recommendations