Document Open Access Logo

Popular Edges with Critical Nodes

Authors Kushagra Chatterjee, Prajakta Nimbhorkar

PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2022.54.pdf
  • Filesize: 0.69 MB
  • 14 pages

Document Identifiers

Author Details

Kushagra Chatterjee
  • National University of Singapore, Singapore
Prajakta Nimbhorkar
  • Chennai Mathematical Institute, India

Funding

  • Chatterjee, Kushagra: Supported in part by an MoE AcRF Tier 2 grant (WBS No. A-8000416-00-00) and an ODPRT grant (WBS No. A-0008078-00-00).
  • Nimbhorkar, Prajakta: Supported in part by SERB Award CRG/2019/004757.

Cite As Get BibTex

Kushagra Chatterjee and Prajakta Nimbhorkar. Popular Edges with Critical Nodes. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ISAAC.2022.54

Abstract

In the popular edge problem, the input is a bipartite graph G = (A ∪ B,E) where A and B denote a set of men and a set of women respectively, and each vertex in A∪ B has a strict preference ordering over its neighbours. A matching M in G is said to be popular if there is no other matching M' such that the number of vertices that prefer M' to M is more than the number of vertices that prefer M to M'. The goal is to determine, whether a given edge e belongs to some popular matching in G. A polynomial-time algorithm for this problem appears in [Cseh and Kavitha, 2018].
We consider the popular edge problem when some men or women are prioritized or critical. A matching that matches all the critical nodes is termed as a feasible matching. It follows from [Telikepalli Kavitha, 2014; Kavitha, 2021; Nasre et al., 2021; Meghana Nasre and Prajakta Nimbhorkar, 2017] that, when G admits a feasible matching, there always exists a matching that is popular among all feasible matchings. 
We give a polynomial-time algorithm for the popular edge problem in the presence of critical men or women. We also show that an analogous result does not hold in the many-to-one setting, which is known as the Hospital-Residents Problem in literature, even when there are no critical nodes.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Matching
  • Stable Matching
  • Popular feasible Matching

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. David J Abraham, Robert W Irving, Telikepalli Kavitha, and Kurt Mehlhorn. Popular matchings. SIAM Journal on Computing, 37(4):1030-1045, 2007. Google Scholar
  2. Péter Biró, Tamás Fleiner, Robert W Irving, and David F Manlove. The college admissions problem with lower and common quotas. Theoretical Computer Science, 411(34-36):3136-3153, 2010. Google Scholar
  3. Kushagra Chatterjee and Prajakta Nimbhorkar. Popular edges with critical nodes. Arxiv Version, 2022. Google Scholar
  4. Ágnes Cseh, Chien-Chung Huang, and Telikepalli Kavitha. Popular matchings with two-sided preferences and one-sided ties. SIAM J. Discret. Math., 31(4):2348-2377, 2017. Google Scholar
  5. Ágnes Cseh and Telikepalli Kavitha. Popular edges and dominant matchings. Mathematical Programming, 172(1):209-229, 2018. Google Scholar
  6. Yuri Faenza and Telikepalli Kavitha. Quasi-popular matchings, optimality, and extended formulations. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, pages 325-344. SIAM, 2020. Google Scholar
  7. Yuri Faenza, Telikepalli Kavitha, Vladlena Powers, and Xingyu Zhang. Popular matchings and limits to tractability. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2790-2809. SIAM, 2019. Google Scholar
  8. David Gale and Lloyd Stowell Shapley. College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9-15, 1962. Google Scholar
  9. David Gale and Marilda Sotomayor. Some remarks on the stable matching problem. Discret. Appl. Math., 11(3):223-232, 1985. Google Scholar
  10. Peter Gärdenfors. Match making: assignments based on bilateral preferences. Behavioral Science, 20(3):166-173, 1975. Google Scholar
  11. Kavitha Gopal, Meghana Nasre, Prajakta Nimbhorkar, and T. Pradeep Reddy. Many-to-one popular matchings with two-sided preferences and one-sided ties. In Ding-Zhu Du, Zhenhua Duan, and Cong Tian, editors, Computing and Combinatorics - 25th International Conference, COCOON 2019, volume 11653 of Lecture Notes in Computer Science, pages 193-205. Springer, 2019. Google Scholar
  12. Koki Hamada, Kazuo Iwama, and Shuichi Miyazaki. The hospitals/residents problem with lower quotas. Algorithmica, 74(1):440-465, 2016. Google Scholar
  13. Chien-Chung Huang and Telikepalli Kavitha. Popular matchings in the stable marriage problem. Information and Computation, 222:180-194, 2013. Google Scholar
  14. Chien-Chung Huang and Telikepalli Kavitha. Popularity, mixed matchings, and self-duality. Math. Oper. Res., 46(2):405-427, 2021. Google Scholar
  15. Telikepalli Kavitha. A size-popularity tradeoff in the stable marriage problem. SIAM J. Comput., 43(1):52-71, 2014. Google Scholar
  16. Telikepalli Kavitha. Popular half-integral matchings. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, volume 55 of LIPIcs, pages 22:1-22:13, 2016. Google Scholar
  17. Telikepalli Kavitha. Popular matchings of desired size. In Graph-Theoretic Concepts in Computer Science - 44th International Workshop, WG 2018, volume 11159 of Lecture Notes in Computer Science, pages 306-317. Springer, 2018. Google Scholar
  18. Telikepalli Kavitha. Min-cost popular matchings. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2020, volume 182 of LIPIcs, pages 25:1-25:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  19. Telikepalli Kavitha. Popular matchings with one-sided bias. In 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, volume 168 of LIPIcs, pages 70:1-70:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  20. Telikepalli Kavitha. Matchings, Critical Nodes, and Popular Solutions. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021), volume 213, pages 25:1-25:19, 2021. Google Scholar
  21. Telikepalli Kavitha. Maximum matchings and popularity. In 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, volume 198 of LIPIcs, pages 85:1-85:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  22. Telikepalli Kavitha, Tamás Király, Jannik Matuschke, Ildikó Schlotter, and Ulrike Schmidt-Kraepelin. Popular branchings and their dual certificates. In Integer Programming and Combinatorial Optimization - 21st International Conference, IPCO 2020, volume 12125 of Lecture Notes in Computer Science, pages 223-237. Springer, 2020. Google Scholar
  23. Donald Ervin Knuth. Marriages stables. Technical report, 1976. Google Scholar
  24. Meghana Nasre and Prajakta Nimbhorkar. Popular matchings with lower quotas. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2017, volume 93 of LIPIcs, pages 44:1-44:15, 2017. Google Scholar
  25. Meghana Nasre, Prajakta Nimbhorkar, Keshav Ranjan, and Ankita Sarkar. Popular Matchings in the Hospital-Residents Problem with Two-Sided Lower Quotas. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021), volume 213, pages 30:1-30:21, 2021. Google Scholar
  26. A. Robards, Paul. Applying two-sided matching processes to the united states navy enlisted assignment process, 2001. Google Scholar
  27. Alvin E. Roth. The evolution of the labor market for medical interns and residents: A case study in game theory. Journal of Political Economy, 92(6):991-1016, 1984. URL: https://doi.org/10.1086/261272.
  28. Alvin E. Roth. On the allocation of residents to rural hospitals: A general property of two-sided matching markets. Econometrica, 54(2):425-427, 1986. Google Scholar
  29. Mallory Soldner. Optimization and measurement in humanitarian operations, 2014. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact