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# Min-Cost Popular Matchings

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LIPIcs.FSTTCS.2020.25.pdf
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## Acknowledgements

Thanks to Jannik Matuschke for conversations on semi-popular matchings and to Piyush Srivastava for helpful discussions on sampling matchings. Thanks to the reviewers for their helpful comments.

## Cite As

Telikepalli Kavitha. Min-Cost Popular Matchings. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.FSTTCS.2020.25

## Abstract

Let G = (A ∪ B, E) be a bipartite graph on n vertices where every vertex ranks its neighbors in a strict order of preference. A matching M in G is popular if there is no matching N such that vertices that prefer N to M outnumber those that prefer M to N. Popular matchings always exist in G since every stable matching is popular. Thus it is easy to find a popular matching in G - however it is NP-hard to compute a min-cost popular matching in G when there is a cost function on the edge set; moreover it is NP-hard to approximate this to any multiplicative factor. An O^*(2ⁿ) algorithm to compute a min-cost popular matching in G follows from known results. Here we show: - an algorithm with running time O^*(2^{n/4}) ≈ O^*(1.19ⁿ) to compute a min-cost popular matching; - assume all edge costs are non-negative - then given ε > 0, a randomized algorithm with running time poly(n,1/(ε)) to compute a matching M such that cost(M) is at most twice the optimal cost and with high probability, the fraction of all matchings more popular than M is at most 1/2+ε.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### Keywords
• Bipartite graphs
• Stable matchings
• Dual certificates

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## References

1. D. J. Abraham, R. W. Irving, T. Kavitha, and K. Mehlhorn. Popular matchings. SIAM Journal on Computing, 37(4):1030-1045, 2007.
2. P. Biro, R. W. Irving, and D. F. Manlove. Popular matchings in the marriage and roommates problems. In Proceedings of the seventh International Conference on Algorithms and Complexity (CIAC), pages 97-108, 2010.
3. 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.
4. Condorcet method. URL: https://en.wikipedia.org/wiki/Condorcet_method.
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. Á. Cseh and T. Kavitha. Popular edges and dominant matchings. Mathematical Programming, 172(1):209-229, 2018.
8. Y. Faenza and T. Kavitha. Quasi-popular matchings, optimality, and extended formulations. In Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 325-344, 2020.
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. T. Feder. A new fixed point approach for stable networks and stable marriages. Journal of Computer and System Sciences, 45(2):233-284, 1992.
11. T. Feder. Network flow and 2-satisfiability. Algorithmica, 11(3):291-319, 1994.
12. T. Fleiner. A fixed-point approach to stable matchings and some applications. Mathematics of Operations Research, 28(1):103-126, 2003.
13. F. V. Fomin and D. Kratsch. Exact exponential algorithms. Springer-Verlag New York, Inc., New York, 2010.
14. D. Gale and L.S. Shapley. College admissions and the stability of marriage. American Mathematical Monthly, 69(1):9-15, 1962.
15. D. Gale and M. Sotomayor. Some remarks on the stable matching problem. Discrete Applied Mathematics, 11:223-232, 1985.
16. P. Gärdenfors. Match making: assignments based on bilateral preferences. Behavioural Science, 20:166-173, 1975.
17. S. Gupta, P. Misra, S. Saurabh, and M. Zehavi. Popular matching in roommates setting is np-hard. In Proceedings of the 30th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2810-2822, 2019.
18. 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.
19. C.-C. Huang and T. Kavitha. Popular matchings in the stable marriage problem. Information and Computation, 222:180-194, 2013.
20. C.-C. Huang and T. Kavitha. Popularity, mixed matchings, and self-duality. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2294-2310, 2017.
21. R. W. Irving, P. Leather, and D. Gusfield. An efficient algorithm for the "optimal" stable marriage. Journal of the ACM, 34(3):532-543, 1987.
22. M. Jerrum and A. Sinclair. Approximating the permanent. SIAM Journal on Computing, 18(6):1149-1178, 1989.
23. T. Kavitha. A size-popularity tradeoff in the stable marriage problem. SIAM Journal on Computing, 43(1):52-71, 2014.
24. 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.
25. T. Kavitha. Popular roommates in simply exponential time. In Proceedings of the 39th Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pages 20:1-20:15, 2019.
26. T. Kavitha, J. Mestre, and M. Nasre. Popular mixed matchings. Theoretical Computer Science, 412:2679-2690, 2011.
27. K. Makarychev, Y. Makarychev, M. Sviridenko, and J. Ward. A bi-criteria approximation algorithm for k-means. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), pages 14:1-14:20, 2016.
28. E. McDermid and R. W. Irving. Sex-equal stable matchings: Complexity and exact algorithms. Algorithmica, 68(3):545-570, 2014.
29. U. G. Rothblum. Characterization of stable matchings as extreme points of a polytope. Mathematical Programming, 54:57-67, 1992.
30. C.-P. Teo and J. Sethuraman. The geometry of fractional stable matchings and its applications. Mathematics of Operations Research, 23(4):874-891, 1998.
31. J. H. Vande Vate. Linear programming brings marital bliss. Operations Research Letters, 8(3):147-153, 1989.
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