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# Popular Edges with Critical Nodes

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LIPIcs.ISAAC.2022.54.pdf
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## Cite As

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

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## References

1. David J Abraham, Robert W Irving, Telikepalli Kavitha, and Kurt Mehlhorn. Popular matchings. SIAM Journal on Computing, 37(4):1030-1045, 2007.
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.
3. Kushagra Chatterjee and Prajakta Nimbhorkar. Popular edges with critical nodes. Arxiv Version, 2022.
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.
5. Ágnes Cseh and Telikepalli Kavitha. Popular edges and dominant matchings. Mathematical Programming, 172(1):209-229, 2018.
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.
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.
8. David Gale and Lloyd Stowell Shapley. College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9-15, 1962.
9. David Gale and Marilda Sotomayor. Some remarks on the stable matching problem. Discret. Appl. Math., 11(3):223-232, 1985.
10. Peter Gärdenfors. Match making: assignments based on bilateral preferences. Behavioral Science, 20(3):166-173, 1975.
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.
12. Koki Hamada, Kazuo Iwama, and Shuichi Miyazaki. The hospitals/residents problem with lower quotas. Algorithmica, 74(1):440-465, 2016.
13. Chien-Chung Huang and Telikepalli Kavitha. Popular matchings in the stable marriage problem. Information and Computation, 222:180-194, 2013.
14. Chien-Chung Huang and Telikepalli Kavitha. Popularity, mixed matchings, and self-duality. Math. Oper. Res., 46(2):405-427, 2021.
15. Telikepalli Kavitha. A size-popularity tradeoff in the stable marriage problem. SIAM J. Comput., 43(1):52-71, 2014.
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.
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.
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.
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.
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.
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.
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.
23. Donald Ervin Knuth. Marriages stables. Technical report, 1976.
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.
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.
26. A. Robards, Paul. Applying two-sided matching processes to the united states navy enlisted assignment process, 2001.
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.
29. Mallory Soldner. Optimization and measurement in humanitarian operations, 2014.
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