Advertisement

Algorithmica

pp 1–34 | Cite as

Subquadratic Algorithms for Succinct Stable Matching

  • Marvin Künnemann
  • Daniel Moeller
  • Ramamohan Paturi
  • Stefan SchneiderEmail author
Article
  • 14 Downloads

Abstract

We consider the stable matching problem when the preference lists are not given explicitly but are represented in a succinct way and ask whether the problem becomes computationally easier and investigate other implications. We give subquadratic algorithms for finding a stable matching in special cases of natural succinct representations of the problem, the d-attribute, d-list, geometric, and single-peaked models. We also present algorithms for verifying a stable matching in the same models. We further show that for \(d = \omega (\log n)\) both finding and verifying a stable matching in the d-attribute and d-dimensional geometric models requires quadratic time assuming the Strong Exponential Time Hypothesis. This suggests that these succinct models are not significantly simpler computationally than the general case for sufficiently large d.

Keywords

Stable matching Attribute model Subquadratic algorithms Conditional lower bound SETH Single-peaked preferences 

Notes

Acknowledgements

We would like to thank Russell Impagliazzo, Vijay Vazirani, and the anonymous reviewers for helpful discussions and comments.

References

  1. 1.
    Abboud, A., Backurs, A., Williams, V.V.: Quadratic-time hardness of LCS and other sequence similarity measures. In: 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS). IEEE (2015)Google Scholar
  2. 2.
    Agarwal, P.K., Arge, L., Erickson, J., Franciosa, P.G., Vitter, J.S.: Efficient searching with linear constraints. In: Proceedings of the Seventeenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, pp. 169–178. ACM (1998)Google Scholar
  3. 3.
    Agarwal, P.K., Erickson, J., et al.: Geometric range searching and its relatives. Contemp. Math. 223, 1–56 (1999)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Alman, J., Williams, R.: Probabilistic polynomials and hamming nearest neighbors. In: 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS). IEEE (2015)Google Scholar
  5. 5.
    Arkin, E.M., Bae, S.W., Efrat, A., Okamoto, K., Mitchell, J.S., Polishchuk, V.: Geometric stable roommates. Inf. Process. Lett. 109(4), 219–224 (2009)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Backurs, A., Indyk, P.: Edit distance cannot be computed in strongly subquadratic time (unless SETH is false). In: Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pp. 51–58 (2015).  https://doi.org/10.1145/2746539.2746612
  7. 7.
    Bartholdi, J., Trick, M.A.: Stable matching with preferences derived from a psychological model. Oper. Res. Lett. 5(4), 165–169 (1986)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bhatnagar, N., Greenberg, S., Randall, D.: Sampling stable marriages: why spouse-swapping won’t work. In: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1223–1232. Society for Industrial and Applied Mathematics (2008)Google Scholar
  9. 9.
    Bogomolnaia, A., Laslier, J.F.: Euclidean preferences. J. Math. Econ. 43(2), 87–98 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bringmann, K.: Why walking the dog takes time: Fréchet distance has no strongly subquadratic algorithms unless seth fails. In: 2014 IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS), pp. 661–670. IEEE (2014)Google Scholar
  11. 11.
    Carmosino, M.L., Gao, J., Impagliazzo, R., Mihajlin, I., Paturi, R., Schneider, S.: Nondeterministic extensions of the strong exponential time hypothesis and consequences for non-reducibility. In: Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, pp. 261–270. ACM (2016)Google Scholar
  12. 12.
    Chebolu, P., Goldberg, L.A., Martin, R.: The complexity of approximately counting stable matchings. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pp. 81–94. Springer (2010)Google Scholar
  13. 13.
    Chung, K.S.: On the existence of stable roommate matchings. Games Econ. Behav. 33(2), 206–230 (2000)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Dabney, J., Dean, B.C.: Adaptive stable marriage algorithms. In: Proceedings of the 48th Annual Southeast Regional Conference, p. 35. ACM (2010)Google Scholar
  15. 15.
    Davis, S., Impagliazzo, R.: Models of greedy algorithms for graph problems. Algorithmica 54(3), 269–317 (2009).  https://doi.org/10.1007/s00453-007-9124-4 MathSciNetzbMATHGoogle Scholar
  16. 16.
    Dobkin, D.P., Kirkpatrick, D.G.: A linear algorithm for determining the separation of convex polyhedra. J. Algorithms 6(3), 381–392 (1985)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69(1), 9–15 (1962)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Gale, D., Sotomayor, M.: Ms. Machiavelli and the stable matching problem. Am. Math. Mon. 92(4), 261–268 (1985)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gonczarowski, Y.A., Nisan, N., Ostrovsky, R., Rosenbaum, W.: A stable marriage requires communication. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1003–1017. SIAM (2015)Google Scholar
  20. 20.
    Gusfield, D.: Three fast algorithms for four problems in stable marriage. SIAM J. Comput. 16(1), 111–128 (1987)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. Foundations of Computing Series. MIT Press, Cambridge (1989)zbMATHGoogle Scholar
  22. 22.
    Hershberger, J., Suri, S.: A pedestrian approach to ray shooting: shoot a ray, take a walk. J. Algorithms 18(3), 403–431 (1995)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Impagliazzo, R., Paturi, R.: On the complexity of k-sat. J. Comput. Syst. Sci. 62(2), 367–375 (2001)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Impagliazzo, R., Paturi, R., Schneider, S.: A satisfiability algorithm for sparse depth two threshold circuits. In: 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS), pp. 479–488. IEEE (2013)Google Scholar
  25. 25.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63, 512–530 (2001)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Irving, R.W.: An efficient algorithm for the “stable roommates” problem. J. Algorithms 6(4), 577–595 (1985)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Irving, R.W., Leather, P.: The complexity of counting stable marriages. SIAM J. Comput. 15(3), 655–667 (1986)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Khuller, S., Mitchell, S.G., Vazirani, V.V.: On-line algorithms for weighted bipartite matching and stable marriages. Theor. Comput. Sci. 127(2), 255–267 (1994)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Knuth, D.E.: Stable Marriage and Its Relation to Other Combinatorial Problems: An Introduction to the Mathematical analysis of Algorithms, vol. 10. American Mathematical Society, Providence (1997)Google Scholar
  30. 30.
    Kobayashi, H., Matsui, T.: Cheating strategies for the gale-shapley algorithm with complete preference lists. Algorithmica 58(1), 151–169 (2010)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Matoušek, J.: Efficient partition trees. Discrete Comput. Geom. 8(1), 315–334 (1992)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Matoušek, J., Schwarzkopf, O.: Linear optimization queries. In: Proceedings of the Eighth Annual Symposium on Computational Geometry, pp. 16–25. ACM (1992)Google Scholar
  33. 33.
    Ng, C., Hirschberg, D.S.: Lower bounds for the stable marriage problem and its variants. SIAM J. Comput. 19(1), 71–77 (1990)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Patrascu, M., Williams, R.: On the possibility of faster SAT algorithms. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pp. 1065–1075 (2010)Google Scholar
  35. 35.
    Razborov, A.A.: Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Math. Notes 41(4), 333–338 (1987)zbMATHGoogle Scholar
  36. 36.
    Roth, A.E.: The economics of matching: stability and incentives. Math. Oper. Res. 7(4), 617–628 (1982)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Roth, A.E., Sotomayor, M.A.O.: Two-Sided Matching: A Study in Game—Theoretic Modeling and Analysis. Econometric Society Monographs. Cambridge University, Cambridge (1990)zbMATHGoogle Scholar
  38. 38.
    Segal, I.: The communication requirements of social choice rules and supporting budget sets. J. Econ. Theory 136(1), 341–378 (2007)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Shapley, L., Scarf, H.: On cores and indivisibility. J. Math. Econ. 1(1), 23–37 (1974)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Smolensky, R.: Algebraic methods in the theory of lower bounds for boolean circuit complexity. In: Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, pp. 77–82. ACM (1987)Google Scholar
  41. 41.
    Teo, C.P., Sethuraman, J., Tan, W.P.: Gale-shapley stable marriage problem revisited: strategic issues and applications. Manag. Sci. 47(9), 1252–1267 (2001)zbMATHGoogle Scholar
  42. 42.
    Williams, R.: A new algorithm for optimal constraint satisfaction and its implications. In: Automata, Languages and Programming, pp. 1227–1237. Springer (2004)Google Scholar
  43. 43.
    Williams, R.: Faster all-pairs shortest paths via circuit complexity. In: Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pp. 664–673. ACM (2014)Google Scholar
  44. 44.
    Williams, R.: Strong ETH breaks with merlin and arthur: short non-interactive proofs of batch evaluation. In: 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, pp. 2:1–2:17 (2016).  https://doi.org/10.4230/LIPIcs.CCC.2016.2

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.University of CaliforniaSan DiegoUSA

Personalised recommendations