Stable Roommate with Narcissistic, Single-Peaked, and Single-Crossing Preferences

  • Robert Bredereck
  • Jiehua ChenEmail author
  • Ugo Paavo Finnendahl
  • Rolf Niedermeier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10576)


The classical Stable Roommate problem asks whether it is possible to pair up an even number of agents such that no two non-paired agents prefer to be with each other rather than with their assigned partners. We investigate Stable Roommate with complete (i.e. every agent can be matched with every other agent) or incomplete preferences, with ties (i.e. two agents are considered of equal value to some agent) or without ties. It is known that in general allowing ties makes the problem NP-complete. We provide algorithms for Stable Roommate that are, compared to those in the literature, more efficient when the input preferences are complete and have some structural property, such as being narcissistic, single-peaked, and single-crossing. However, when the preferences are incomplete and have ties, we show that being single-peaked and single-crossing does not reduce the computational complexity—Stable Roommate remains NP-complete.


  1. 1.
    Ballester, M.Á., Haeringer, G.: A characterization of the single-peaked domain. Soc. Choice Welf. 36(2), 305–322 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barberà, S., Moreno, B.: Top monotonicity: a common root for single peakedness, single crossing and the median voter result. Game Econ. Behav. 73(2), 345–359 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bartholdi, J.J., Trick, M.: Stable matching with preferences derived from a psychological model. Oper. Res. Lett. 5(4), 165–169 (1986)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Black, D.: The Theory of Committees and Elections. Cambridge University Press, Cambridge (1958)zbMATHGoogle Scholar
  5. 5.
    Bredereck, R., Chen, J., Woeginger, G.J.: A characterization of the single-crossing domain. Soc. Choice Welf. 41(4), 989–998 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bredereck, R., Chen, J., Woeginger, G.J.: Are there any nicely structured preference profiles nearby? Math. Soc. Sci. 79, 61–73 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Coombs, C.H.: A Theory of Data. Wiley, New York (1964)Google Scholar
  8. 8.
    Doignon, J., Falmagne, J.: A polynomial time algorithm for unidimensional unfolding representations. J. Algorithms 16(2), 218–233 (1994)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Elkind, E., Faliszewski, P., Slinko, A.: Clone structures in voters’ preferences. In: Proceedings of EC 2012, pp. 496–513 (2012)Google Scholar
  10. 10.
    Elkind, E., Faliszewski, P., Skowron, P.: A characterization of the single-peaked single-crossing domain. In: Proceedings of AAAI 2014, pp. 654–660 (2014)Google Scholar
  11. 11.
    Elkind, E., Faliszewski, P., Lackner, M., Obraztsova, S.: The complexity of recognizing incomplete single-crossing preferences. In: Proceedings of AAAI 2015, pp. 865–871, Long version available as manuscript on M. Lackers homepage (2015)Google Scholar
  12. 12.
    Elkind, E., Lackner, M., Peters, D.: Structured preferences. In: Endriss, U. (ed.) Trends in Computational Social Choice. AI Access (2017). Upcoming. Draft online availableGoogle Scholar
  13. 13.
    Escoffier, B., Lang, J., Öztürk, M.: Single-peaked consistency and its complexity. In: Proceedings of ECAI 2008, pp. 366–370 (2008)Google Scholar
  14. 14.
    Fitzsimmons, Z.: Single-peaked consistency for weak orders is easy. Technical report. arXiv:1406.4829v3 [cs.GT] (2016)
  15. 15.
    Fitzsimmons, Z., Hemaspaandra, E.: Modeling single-peakedness for votes with ties. In: Proceedings of STAIRS 2016, vol. 284. Frontiers in Artificial Intelligence and Applications, pp. 63–74 (2016)Google Scholar
  16. 16.
    Gai, A.-T., Lebedev, D., Mathieu, F., Montgolfier, F., Reynier, J., Viennot, L.: Acyclic preference systems in P2P networks. In: Kermarrec, A.-M., Bougé, L., Priol, T. (eds.) Euro-Par 2007. LNCS, vol. 4641, pp. 825–834. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-74466-5_88 CrossRefGoogle Scholar
  17. 17.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 120(5), 386–391 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Garey, M.R., Johnson, D.S.: Computers and Intractability–A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  19. 19.
    Gusfield, D., Irving, R.W.: The Stable Marriage Problem-Structure and Algorithms. Foundations of Computing Series. MIT Press, Cambridge (1989)zbMATHGoogle Scholar
  20. 20.
    Hotelling, H.: Stability in competition. Econ. J. 39(153), 41–57 (1929)CrossRefGoogle Scholar
  21. 21.
    Irving, R.W.: An efficient algorithm for the “stable roommates” problem. J. Algorithms 6(4), 577–595 (1985)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Irving, R.W., Manlove, D.: The stable roommates problem with ties. J. Algorithms 43(1), 85–105 (2002)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Knuth, D.E.: Stable marriage and its relation to other combinatorial problems. CRM Proceedings & Lecture Notes, vol. 10. AMS (1997)Google Scholar
  24. 24.
    Kujansuu, E., Lindberg, T., Mäkinen, E.: The stable roommates problem and chess tournament pairings. Divulgaciones Matemáticas 7, 19–28 (1999)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Lackner, M.: Incomplete preferences in single-peaked electorates. In: Proceedings of AAAI 2014, pp. 742–748 (2014)Google Scholar
  26. 26.
    Lebedev, D., Mathieu, F., Viennot, L., Gai, A., Reynier, J., de Montgolfier, F.: On using matching theory to understand P2P network design. Technical report. arXiv:cs/0612108v1 [cs.NI] (2006)
  27. 27.
    Manlove, D.F.: Algorithmics of Matching Under Preferences. Series on Theoretical Computer Science, vol. 2. WorldScientific, Singapore (2013)CrossRefGoogle Scholar
  28. 28.
    Manlove, D.F., O’Malley, G.: Paired and altruistic kidney donation in the UK: algorithms and experimentation. ACM J. Exp. Algorithmics 19(1), 2.6:1–2.6:21 (2014)MathSciNetGoogle Scholar
  29. 29.
    Mirrlees, J.A.: An exploration in the theory of optimal income taxation. Rev. Econ. Stud. 38, 175–208 (1971)CrossRefGoogle Scholar
  30. 30.
    Roberts, K.W.: Voting over income tax schedules. J. Public Econ. 8(3), 329–340 (1977)CrossRefGoogle Scholar
  31. 31.
    Ronn, E.: NP-complete stable matching problems. J. Algorithms 11(2), 285–304 (1990)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Roth, A.E., Sotomayor, M., Matching, T.-S.: A Study in Game-Theoretic Modeling and Analysis. Cambridge University Press, Cambridge (1990)Google Scholar
  33. 33.
    Roth, A.E., Sönmez, T., Ünver, M.U.: Pairwise kidney exchange. J. Econ. Theor. 125(2), 151–188 (2005)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Roth, A.E., Sönmez, T., Ünver, M.U.: Efficient kidney exchange: coincidence of wants in markets with compatibility-based preferences. Am. Econ. Rev. 97(3), 828–851 (2007)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Robert Bredereck
    • 1
  • Jiehua Chen
    • 1
    • 2
    Email author
  • Ugo Paavo Finnendahl
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.TU BerlinBerlinGermany
  2. 2.Ben-Gurion University of the NegevBeershebaIsrael

Personalised recommendations