Document

# Stable Matchings with One-Sided Ties and Approximate Popularity

## File

LIPIcs.FSTTCS.2022.22.pdf
• Filesize: 0.71 MB
• 17 pages

## Acknowledgements

Thanks to the reviewers for their helpful comments and suggestions.

## Cite As

Telikepalli Kavitha. Stable Matchings with One-Sided Ties and Approximate Popularity. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.FSTTCS.2022.22

## Abstract

We consider a matching problem in a bipartite graph G = (A ∪ B, E) where vertices in A rank their neighbors in a strict order of preference while vertices in B are allowed to have weak rankings, i.e., ties are allowed in their preferences. Stable matchings always exist in G and are easy to find, however popular matchings need not exist and it is NP-complete to decide if one exists. This motivates the "approximately popular" matching problem. A well-known measure of approximate popularity is low unpopularity factor. We show that when each tie in G has length at most k, there always exists a stable matching whose unpopularity factor is at most k. Our proof is algorithmic and we compute such a stable matching in polynomial time. Our result can be considered to be a generalization of Gärdenfors' result (1975) which showed that when rankings are strict, every stable matching is popular. There are several applications where the size of the matching is its most important attribute. What one seeks here is a maximum matching M such that there is no maximum matching more popular than M. When rankings are weak, it is NP-hard to decide if G admits such a matching. When ties are one-sided and of length at most k, we show a polynomial time algorithm to find a maximum matching whose unpopularity factor within the set of maximum matchings is at most 2k.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### Keywords
• Bipartite graphs
• Maximum matchings
• Unpopularity factor

## Metrics

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

## References

1. F. Bauckholt, K. Pashkovich, and L. Sanitá. On the approximability of the stable marriage problem with one-sided ties. URL: http://arxiv.org/abs/1805.05391.
2. S. Bhattacharya, M. Hoefer, C.-C. Huang, T. Kavitha, and L. Wagner. Maintaining near-popular matchings. In Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming (ICALP)(II), pages 504-515, 2015.
3. P. Biro, R. W. Irving, and D. F. Manlove. Popular matchings in the marriage and roommates problems. In Proceedings of the 7th International Conference on Algorithms and Complexity (CIAC), pages 97-108, 2010.
4. P. Biro, D. F. Manlove, and S. Mittal. Size versus stability in the marriage problem. Theoretical Computer Science, 411:1828-1841, 2010.
5. Á. Cseh. Popular matchings. Trends in Computational Social Choice, Ulle Endriss (ed.), 2017.
6. Á. Cseh, C.-C. Huang, and T. Kavitha. Popular matchings with two-sided preferences and one-sided ties. SIAM Journal on Discrete Mathematics, 31(4):2348-2377, 2017.
7. A. L. Dulmage and N. S. Mendelsohn. Coverings of bipartite graphs. Canadian Journal of Mathematics, 10:517-534, 1958.
8. Y. Faenza and T. Kavitha. Quasi-popular matchings, optimality, and extended formulations. Mathematics of Operations Research, 47(1):427-457, 2022.
9. Y. Faenza, T. Kavitha, V. Powers, and X. Zhang. Popular matchings and limits to tractability. In Proceedings of the 30th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2790-2809, 2019.
10. D. Gale and L.S. Shapley. College admissions and the stability of marriage. American Mathematical Monthly, 69(1):9-15, 1962.
11. P. Gärdenfors. Match making: assignments based on bilateral preferences. Behavioural Science, 20:166-173, 1975.
12. S. Gupta, P. Misra, S. Saurabh, and M. Zehavi. Popular matching in roommates setting is NP-hard. ACM Transactions on Computation Theory, 13(2):1-20, 2021.
13. C.-C. Huang and T. Kavitha. Near-popular matchings in the roommates problem. SIAM Journal on Discrete Mathematics, 27(1):43-62, 2013.
14. C.-C. Huang and T. Kavitha. Improved approximation algorithms for two variants of the stable marriage problem with ties. Mathematical Programming, 154(1):353-380, 2015.
15. K. Iwama, S. Miyazaki, and N. Yamauchi. A 1.875-approximation algorithm for the stable marriage problem. In Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 288-297, 2007.
16. K. Iwama, S. Miyazaki, and N. Yamauchi. A 25/17-approximation algorithm for the stable marriage problem with one-sided ties. Algorithmica, 68(3):758-775, 2014.
17. T. Kavitha. A size-popularity tradeoff in the stable marriage problem. SIAM Journal on Computing, 43(1):52-71, 2014.
18. Z. Király. Better and simpler approximation algorithms for the stable marriage problem. Algorithmica, 60(1):3-20, 2011.
19. C.-K. Lam and C. G. Plaxton. A (1+1/e)-approximation algorithm for maximum stable matching with one-sided ties and incomplete lists. In Proceedings of the 30th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2823-2840, 2019.
20. L. Losász and M. D. Plummer. Matching theory. North-Holland, Mathematics Studies 121, 1986.
21. D. F. Manlove and R. W. Irving. Approximation algorithms for hard variants of the stable marriage and hospitals/residents problems. Journal of Combinatorial Optimization, 16:279-292, 2008.
22. M. McCutchen. The least-unpopularity-factor and least-unpopularity-margin criteria for matching problems with one-sided preferences. In Proceedings of the 8th Latin American Symposium on Theoretical Informatics (LATIN), pages 593-604, 2008.
23. W. R. Pulleyblank. Chapter 3, matchings and extensions. The Handbook of Combinatorics, R.L. Graham, M. Grötschel, and L. Lovasz (ed.), 1995.
24. S. Ruangwises and T. Itoh. Unpopularity factor in the marriage and roommates problems. Theory of Computing Systems, 65(3):579-592, 2021.
25. M. Soldner. Optimization and measurement in humanitarian operations: Addressing practical needs. PhD thesis, Georgia Institute of Technology, 2014.
26. A.C. Trapp, A. Teytelboym, A. Martinello, T. Andersson, and N. Ahani. Placement optimization in refugee resettlement. Working paper, 2018.