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Matchings, Critical Nodes, and Popular Solutions

Author Telikepalli Kavitha



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Author Details

Telikepalli Kavitha
  • Tata Institute of Fundamental Research, Mumbai, India

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Telikepalli Kavitha. Matchings, Critical Nodes, and Popular Solutions. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 25:1-25:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.FSTTCS.2021.25

Abstract

We consider a matching problem in a marriage instance G. Every node has a strict preference order ranking its neighbors. There is a set C of prioritized or critical nodes and we are interested in only those matchings that match as many critical nodes as possible. Such matchings are useful in several applications and we call them critical matchings. A stable matching need not be critical. We consider a well-studied relaxation of stability called popularity. Our goal is to find a popular critical matching, i.e., a weak Condorcet winner within the set of critical matchings where nodes are voters. We show that popular critical matchings always exist in G and min-size/max-size such matchings can be efficiently computed.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Bipartite graphs
  • Stable matchings
  • LP-duality

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References

  1. A. Abdulkadiroğlu and T. Sönmez. School choice: a mechanism design approach. American Economic Review, 93(3):729-747, 2003. Google Scholar
  2. S. Baswana, P. P. Chakrabarti, S. Chandran, Y. Kanoria, and U. Patange. Centralized admissions for engineering colleges in India. INFORMS Journal on Applied Analytics, 49(5):338-354, 2019. Google Scholar
  3. P. Biro, D. F. Manlove, and S. Mittal. Size versus stability in the marriage problem. Theoretical Computer Science, 411:1828-1841, 2010. Google Scholar
  4. Canadian Resident Matching Service. How the matching algorithm works. URL: http://carms.ca/algorithm.htm.
  5. M.-J.-A.-N. de C. (Marquis de) Condorcet. Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix. L'Imprimerie Royale, 1785. Google Scholar
  6. Á. Cseh. Popular matchings. Trends in Computational Social Choice, Ulle Endriss (ed.), 2017. Google Scholar
  7. Á. Cseh and T. Kavitha. Popular edges and dominant matchings. Mathematical Programming, 172(1):209-229, 2018. Google Scholar
  8. K. Eriksson and O. Häggström. Instability of matchings in decentralized markets with various preference orders. Mathematical Programming, 36(3-4):409-420, 2008. Google Scholar
  9. D. Gale and L.S. Shapley. College admissions and the stability of marriage. American Mathematical Monthly, 69(1):9-15, 1962. Google Scholar
  10. D. Gale and M. Sotomayor. Some remarks on the stable matching problem. Discrete Applied Mathematics, 11(3):223-232, 1985. Google Scholar
  11. P. Gärdenfors. Match making: assignments based on bilateral preferences. Behavioural Science, 20:166-173, 1975. Google Scholar
  12. M. Hirakawa, Y. Yamauchi, S. Kijima, and M. Yamashita. On the structure of popular matchings in the stable marriage problem: Who can join a popular matching? In the 3rd International Workshop on Matching Under Preferences (MATCH-UP), 2015. Google Scholar
  13. C.-C. Huang and T. Kavitha. Popular matchings in the stable marriage problem. Information and Computation, 222:180-194, 2013. Google Scholar
  14. T. Kavitha. A size-popularity tradeoff in the stable marriage problem. SIAM Journal on Computing, 43(1):52-71, 2014. Google Scholar
  15. T. Kavitha. Popular half-integral matchings. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP), pages 22:1-22:13, 2016. Google Scholar
  16. T. Kavitha. Maximum matchings and popularity. In Proceedings of the 48th International Colloquium on Automata, Languages, and Programming (ICALP), pages 85:1-85:21, 2021. Google Scholar
  17. C. Mathieu. Stable matching in practice. In the 26th Annual European Symposium on Algorithms (ESA), Keynote talk, 2018. Google Scholar
  18. S. Merrill and B. Grofman. A unified theory of voting: directional and proximity spatial models. Cambridge University Press, 1999. Google Scholar
  19. M. Nasre and P. Nimbhorkar. Popular matchings with lower quotas. In Proceedings of the 37th Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pages 44:1-44:15, 2017. Google Scholar
  20. M. Nasre, P. Nimbhorkar, K. Ranjan, and A. Sarkar. Popular matchings in the hospitals-residents problem with two-sided lower quotas. In Proceedings of the 41st Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 2021. Google Scholar
  21. National Resident Matching Program. Why the Match? URL: http://www.nrmp.org/whythematch.pdf.
  22. P. A. Robards. Applying the two-sided matching processes to the United States Navy enlisted assignment process. Master’s Thesis, Naval Postgraduate School, Monterey, Canada, 2001. Google Scholar
  23. A. E. Roth and X. Xing. Turnaround time and bottlenecks in market clearing: Decentralized matching in the market for clinical psychologists. Journal of Political Economy, 105(2):284-329, 1997. Google Scholar
  24. M. Soldner. Optimization and measurement in humanitarian operations: Addressing practical needs. PhD thesis, Georgia Institute of Technology, 2014. Google Scholar
  25. A.C. Trapp, A. Teytelboym, A. Martinello, T. Andersson, and N. Ahani. Placement optimization in refugee resettlement. Working paper, 2018. Google Scholar
  26. W. Yang, J. A. Giampapa, and K. Sycara. Two-sided matching for the US Navy detailing process with market complication. Technical Report CMU-R1-TR-03-49, Robotics Institute, Carnegie Mellon University, 2003. Google Scholar
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