Satisfying Neighbor Preferences on a Circle

  • Danny Krizanc
  • Manuel Lafond
  • Lata Narayanan
  • Jaroslav Opatrny
  • Sunil Shende
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


We study the problem of satisfying seating preferences on a circle. We assume we are given a collection of n agents to be arranged on a circle. Each agent is colored either blue or red, and there are exactly b blue agents and r red agents. The w-neighborhood of an agent A is the sequence of \(2w+1\) agents at distance \(\le \) \(w\) from A in the clockwise circular ordering. Agents have preferences for the colors of other agents in their w-neighborhood. We consider three ways in which agents can express their preferences: each agent can specify (1) a preference list: the sequence of colors of agents in the neighborhood, (2) a preference type: the exact number of neighbors of its own color in its neighborhood, or (3) a preference threshold: the minimum number of agents of its own color in its neighborhood. Our main result is that satisfying seating preferences is fixed-parameter tractable (FPT) with respect to parameter w for preference types and thresholds, while it can be solved in O(n) time for preference lists. For some cases of preference types and thresholds, we give O(n) algorithms whose running time is independent of w.


Seating arrangement Linear algorithm FPT algorithm 


  1. 1.
    Andrews, J.A., Jacobson, M.S.: On a generalization of chromatic number. Congr. Numer. 47, 33–48 (1985)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Benard, S., Willer, R.: A wealth and status-based model of residential segregation. Math. Sociol. 31(2), 149–174 (2007)CrossRefGoogle Scholar
  3. 3.
    Benenson, I., Hatna, E., Or, E.: From schelling to spatially explicit modeling of urban ethnic and economic residential dynamics. Sociol. Methods Res. 37(4), 463–497 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brandt, C., Immorlica, N., Kamath, G., Kleinberg, R.: An analysis of one-dimensional Schelling segregation. In: Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, pp. 789–804. ACM (2012)Google Scholar
  5. 5.
    Cowen, L.J., Cowen, R.H., Woodall, D.R.: Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valency. J. Gr. Theory 10(2), 187–195 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cowen, L.J., Goddard, W., Jesurum, C.E.: Coloring with defect. In: SODA, pp. 548–557 (1997)Google Scholar
  7. 7.
    Cowen, L.J., Goddard, W., Jesurum, C.E.: Defective coloring revisited. J. Gr. Theory 24(3), 205–219 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dall’Asta, L., Castellano, C., Marsili, M.: Statistical physics of the schelling model of segregation. J. Stat. Mech: Theory Exp. 2008(07), L07002 (2008)Google Scholar
  9. 9.
    Frank, A., Tardos, É.: An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Henry, A.D., Pralat, P., Zhang, C.-Q.: Emergence of segregation in evolving social networks. Proc. Natl. Acad. Sci. 108(21), 8605–8610 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Immorlica, N., Kleinberg, R., Lucier, B., Zadomighaddam, M.: Exponential segregation in a two-dimensional schelling model with tolerant individuals. In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 984–993. SIAM (2017)Google Scholar
  12. 12.
    Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12(3), 415–440 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kuhn, F.: Weak graph colorings: distributed algorithms and applications. In: Proceedings of the Twenty-First Annual Symposium on Parallelism in Algorithms and Architectures, pp. 138–144. ACM (2009)Google Scholar
  14. 14.
    Lenstra Jr., H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lokshtanov, D.: New methods in parameterized algorithms and complexity. Ph.D. thesis. University of Bergen, Norway (2009)Google Scholar
  16. 16.
    Pancs, R., Vriend, N.J.: Schelling’s spatial proximity model of segregation revisited. J. Public Econ. 91(1), 1–24 (2007)CrossRefGoogle Scholar
  17. 17.
    Parshina, O.G.: Perfect 2-colorings of infinite circulant graphs with continuous set of distances. J. Appl. Ind. Math. 8(3), 357–361 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pevzner, P.A., Tang, H., Waterman, M.S.: An Eulerian path approach to DNA fragment assembly. Proc. Natl. Acad. Sci. 98(17), 9748–9753 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Schelling, T.C.: Models of segregation. Am. Econ. Rev. 59(2), 488–493 (1969)Google Scholar
  20. 20.
    Schelling, T.C.: Dynamic models of segregation. J. Math. Sociol. 1(2), 143–186 (1971)CrossRefzbMATHGoogle Scholar
  21. 21.
    Young, H.P.: Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. Princeton University Press, Princeton (2001)Google Scholar
  22. 22.
    Zhang, J.: A dynamic model of residential segregation. J. Math. Sociol. 28(3), 147–170 (2004)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceWesleyan UniversityMiddletownUSA
  2. 2.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  3. 3.Department of Computer Science and Software EngineeringConcordia UniversityMontrealCanada
  4. 4.Department of Computer ScienceRutgers UniversityCamdenUSA

Personalised recommendations