Abstract
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.
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References
Andrews, J.A., Jacobson, M.S.: On a generalization of chromatic number. Congr. Numer. 47, 33–48 (1985)
Benard, S., Willer, R.: A wealth and status-based model of residential segregation. Math. Sociol. 31(2), 149–174 (2007)
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)
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)
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)
Cowen, L.J., Goddard, W., Jesurum, C.E.: Coloring with defect. In: SODA, pp. 548–557 (1997)
Cowen, L.J., Goddard, W., Jesurum, C.E.: Defective coloring revisited. J. Gr. Theory 24(3), 205–219 (1997)
Dall’Asta, L., Castellano, C., Marsili, M.: Statistical physics of the schelling model of segregation. J. Stat. Mech: Theory Exp. 2008(07), L07002 (2008)
Frank, A., Tardos, É.: An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)
Henry, A.D., Pralat, P., Zhang, C.-Q.: Emergence of segregation in evolving social networks. Proc. Natl. Acad. Sci. 108(21), 8605–8610 (2011)
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)
Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12(3), 415–440 (1987)
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)
Lenstra Jr., H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)
Lokshtanov, D.: New methods in parameterized algorithms and complexity. Ph.D. thesis. University of Bergen, Norway (2009)
Pancs, R., Vriend, N.J.: Schelling’s spatial proximity model of segregation revisited. J. Public Econ. 91(1), 1–24 (2007)
Parshina, O.G.: Perfect 2-colorings of infinite circulant graphs with continuous set of distances. J. Appl. Ind. Math. 8(3), 357–361 (2014)
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)
Schelling, T.C.: Models of segregation. Am. Econ. Rev. 59(2), 488–493 (1969)
Schelling, T.C.: Dynamic models of segregation. J. Math. Sociol. 1(2), 143–186 (1971)
Young, H.P.: Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. Princeton University Press, Princeton (2001)
Zhang, J.: A dynamic model of residential segregation. J. Math. Sociol. 28(3), 147–170 (2004)
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Krizanc, D., Lafond, M., Narayanan, L., Opatrny, J., Shende, S. (2018). Satisfying Neighbor Preferences on a Circle. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_53
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DOI: https://doi.org/10.1007/978-3-319-77404-6_53
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