Maximizing Dominance in the Plane and Its Applications

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11646)


Given a set P of n points with weights (possibly negative), a set Q of m points in the plane, and a positive integer k, we consider the optimization problem of finding a subset of Q with at most k points that dominates a subset of P with maximum total weight. We say a set of points \(Q'\) dominates p if some point q of \(Q'\) satisfies \(x(p)\leqslant x(q)\) and \(y(p)\leqslant y(q)\). We present an efficient algorithm solving this problem in \(O(k(n+m)\log m)\) time and \(O(n+m)\) space. Our result implies algorithms with better time bounds for related problems, including the disjoint union of cliques problem for interval graphs (equivalently, the hitting intervals problem) and the top-k representative skyline points problem in the plane.


Dominance Disjoint cliques Hitting intervals 


  1. 1.
    Alrifai, M., Skoutas, D., Risse, T.: Selecting skyline services for QoS-based web service composition. In: Proceedings of the 19th International Conference on World Wide Web, WWW 2010, pp. 11–20. ACM (2010)Google Scholar
  2. 2.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bringmann, K., Cabello, S., Emmerich, M.T.M.: Maximum volume subset selection for anchored boxes. In: 33rd International Symposium on Computational Geometry (SoCG 2017), vol. 77. Leibniz International Proceedings in Informatics (LIPIcs), pp. 22:1–22:15. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2017)Google Scholar
  4. 4.
    Bringmann, K., Friedrich, T., Klitzke, P.: Two-dimensional subset selection for hypervolume and epsilon-indicator. In: GECCO, pp. 589–596. ACM (2014)Google Scholar
  5. 5.
    Chrobak, M., Golin, M., Lam, T.-W., Nogneng, D.: Scheduling with gaps: new models and algorithms. In: Paschos, V.T., Widmayer, P. (eds.) CIAC 2015. LNCS, vol. 9079, pp. 114–126. Springer, Cham (2015). Scholar
  6. 6.
    Damaschke, P.: Refined algorithms for hitting many intervals. Inf. Process. Lett. 118, 117–122 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    de Berg, M., Cheong, O., van Kreveld, M.J., Overmars, M.H.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008). Scholar
  8. 8.
    Ertem, Z., Lykhovyd, E., Wang, Y., Butenko, S.: The maximum independent union of cliques problem: complexity and exact approaches. J. Glob. Optim. (2018)Google Scholar
  9. 9.
    Gavril, F.: Algorithms for maximum k-colorings and k-coverings of transitive graphs. Networks 17(4), 465–470 (1987)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jansen, K., Scheffler, P., Woeginger, G.: The disjoint cliques problem. RAIRO Recherhe Opérationnelle 31, 45–66 (1997)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kuhn, T., Fonseca, C.M., Paquete, L., Ruzika, S., Duarte, M.M., Figueira, J.R.: Hypervolume subset selection in two dimensions: formulations and algorithms. Evol. Comput. 24(3), 411–425 (2016)CrossRefGoogle Scholar
  12. 12.
    Lin, X., Yuan, Y., Zhang, Q., Zhang, Y.: Selecting stars: the \(k\) most representative skyline operator. In: Proceedings of the 23rd International Conference on Data Engineering, ICDE 2007, pp. 86–95 (2007)Google Scholar
  13. 13.
    Tao, Y., Ding, L., Lin, X., Pei, J.: Distance-based representative skyline. In: Proceedings of the 25th International Conference on Data Engineering, ICDE 2009, pp. 892–903. IEEE Computer Society (2009)Google Scholar
  14. 14.
    Yannakakis, M., Gavril, F.: The maximum k-colorable subgraph problem for chordal graphs. Inf. Process. Lett. 24(2), 133–137 (1987)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Yuan, L., Qin, X.L., Chang, L., Zhang, W.: Diversified top-k clique search. VLDB J. 25(2), 171–196 (2016)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringPohang University of Science and TechnologyPohangSouth Korea
  2. 2.Department of Mathematics, IMFMUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Department of Mathematics, FMFUniversity of LjubljanaLjubljanaSlovenia

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