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Popular Half-Integral Matchings

Author Telikepalli Kavitha

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Keywords
  • bipartite graphs
  • stable matchings
  • fractional matchings
  • polytopes

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Abstract

In an instance G = (A union B, E) of the stable marriage problem with strict and possibly incomplete preference lists, a matching M is popular if there is no matching M0 where the vertices that prefer M' to M outnumber those that prefer M to M'. All stable matchings are popular and there is a simple linear time algorithm to compute a maximum-size popular matching. More generally, what we seek is a min-cost popular matching where we assume there is a cost function c : E -> Q. However there is no polynomial time algorithm currently known for solving this problem. Here we consider the following generalization of a popular matching called a popular half-integral matching: this is a fractional matching ~x = (M_1 + M_2)/2, where M1 and M2 are the 0-1 edge incidence vectors of matchings in G, such that ~x satisfies popularity constraints. We show that every popular half-integral matching is equivalent to a stable matching in a larger graph G^*. This allows us to solve the min-cost popular half-integral matching problem in polynomial time.

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Telikepalli Kavitha. Popular Half-Integral Matchings. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 22:1-22:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.ICALP.2016.22

Author Details

Telikepalli Kavitha

References

  1. D. J. Abraham, R. W. Irving, T. Kavitha, and K. Mehlhorn. Popular matchings. SIAM Journal on Computing, 37(4):1030-1045, 2007. Google Scholar
  2. P. Biró, R. W. Irving, and D. F. Manlove. Popular matchings in the marriage and roommates problems. In Proceedings of 7th International Conference on Algorithms and Complexity (CIAC), pages 97-108, 2010. Google Scholar
  3. Á. Cseh, C.-C. Huang, and T. Kavitha. Popular matchings with two-sided preferences and one-sided ties. In Proceedings of 42nd International Colloquium on Automata, Languages, and Programming (ICALP), pages 367-379, 2015. Google Scholar
  4. Á. Cseh and T. Kavitha. Popular edges and dominant matchings. To appear in the Proceedings of the 18th Conference on Integer Programming and Combinatorial Optimization (IPCO), 2016. Google Scholar
  5. T. Feder. A new fixed point approach for stable networks and stable marriages. Journal of Computer and System Sciences, 45:233-284, 1992. Google Scholar
  6. T. Feder. Network flow and 2-satisfiability. Algorithmica, 11(3):291-319, 1994. Google Scholar
  7. D. Gale and L.S. Shapley. College admissions and the stability of marriage. American Mathematical Monthly, 69:9-15, 1962. Google Scholar
  8. D. Gale and M. Sotomayor. Some remarks on the stable matching problem. Discrete Applied Mathematics, 11:223-232, 1985. Google Scholar
  9. P. Gärdenfors. Match making: assignments based on bilateral preferences. Behavioural Sciences, 20:166-173, 1975. Google Scholar
  10. C.-C. Huang and T. Kavitha. Popular matchings in the stable marriage problem. Information and Computation, 222:180-194, 2013. Google Scholar
  11. R. W. Irving, P. Leather, and D. Gusfield. An efficient algorithm for the "optimal" stable marriage. Journal of the ACM, 38(3):532-543, 1987. Google Scholar
  12. T. Kavitha. A size-popularity tradeoff in the stable marriage problem. SIAM Journal on Computing, 43(1):52-71, 2014. Google Scholar
  13. T. Kavitha, J. Mestre, and M. Nasre. Popular mixed matchings. Theoretical Computer Science, 412:2679-2690, 2011. Google Scholar
  14. U. Rothblum. Characterization of stable matchings as extreme points of a polytope. Mathematical Programming, 54:57-67, 1992. Google Scholar
  15. C.-P. Teo and J. Sethuraman. The geometry of fractional stable matchings and its applications. Mathematics of Operations Research, 23(4):874-891, 1998. Google Scholar
  16. J. H. Vande Vate. Linear programming brings marital bliss. Operations Research Letters, 8(3):147-153, 1989. Google Scholar
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