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Maximizing Dominance in the Plane and its Applications

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Abstract

Given a set P of n weighted points, 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. A set \(Q'\) of points in the plane dominates a point p in the plane if some point \(q\in Q'\) 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.

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Notes

  1. Given the amount of interrelated results and possible names for the problem, we hope we are not victims of the same problem.

  2. A data structure with similar guarantees can be constructed using cut-link trees [16], top trees [2], or other dynamic trees. Depending on the actual dynamic tree underlying the construction, the bounds may be amortized.

References

  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)

  2. Alstrup, S., Holm, J., de Lichtenberg, K., Thorup, Mikkel: Maintaining information in fully dynamic trees with top trees. ACM Trans. Algorith. 1(2), 243–264 (2005)

    Article  MathSciNet  Google Scholar 

  3. Booth, Kellogg S., Lueker, George 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)

    Article  MathSciNet  Google Scholar 

  4. Bringmann, K., Cabello, S., Emmerich, M.T.M.: Maximum volume subset selection for anchored boxes. In: Proceedings of the 33rd International Symposium on Computational Geometry (SoCG 2017), volume 77 of Leibniz International Proceedings in Informatics (LIPIcs), pp. 22:1–22:15. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2017)

  5. Bringmann, K., Friedrich, T., Klitzke, P.: Two-dimensional subset selection for hypervolume and epsilon-indicator. In: Proceedings of the Annual Conference on Genetic and Evolutionary Computation (GECCO 2014), pp. 589–596. ACM (2014)

  6. Chan, T.M.: Optimal output-sensitive convex hull algorithms in two and three dimensions. Discret. Comput. Geom. 16(4), 361–368 (1996)

    Article  MathSciNet  Google Scholar 

  7. Chrobak, M., Golin, M.J., Lam, T.W., Nogneng, D.: Scheduling with gaps: new models and algorithms. In: Proceedings of the 9th International Conference on Algorithms and Complexity (CIAC 2015), volume 9079 of Lecture Notes in Computer Science (LNCS), pp. 114–126. Springer (2015)

  8. Damaschke, P.: Refined algorithms for hitting many intervals. Inf. Process. Lett. 118, 117–122 (2017)

    Article  MathSciNet  Google Scholar 

  9. de Berg, M., Cheong, O., van Kreveld, M.J., Overmars, M.H.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer (2008)

  10. Ertem, Z., Lykhovyd, E., Wang, Y., Butenko, S.: The maximum independent union of cliques problem: complexity and exact approaches. J. Glob. Optim., Published online (2018)

  11. Gavril, F.: Algorithms for maximum k-colorings and k-coverings of transitive graphs. Networks 17(4), 465–470 (1987)

    Article  MathSciNet  Google Scholar 

  12. Jansen, K., Scheffler, P., Woeginger, G.: The disjoint cliques problem. RAIRO Recherhe Opérationnelle 31, 45–66 (1997)

    MathSciNet  MATH  Google Scholar 

  13. Kirkpatrick, D.G., Seidel, R.: Output-size sensitive algorithms for finding maximal vectors. In: Proceedings of the First Annual Symposium on Computational Geometry (SoCG 1985), pp. 89–96. ACM (1985)

  14. Kuhn, T., Fonseca, C.M., Paquete, L., Ruzika, S., Duarte, M.M., Figueira, J.: Hypervolume subset selection in two dimensions: formulations and algorithms. Evol. Comput. 24(3), 411–425 (2016)

    Article  Google Scholar 

  15. 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. IEEE Computer Society (2007)

  16. Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 26(3), 362–391 (1983)

    Article  MathSciNet  Google Scholar 

  17. 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)

  18. Yannakakis, M., Gavril, F.: The maximum k-colorable subgraph problem for chordal graphs. Inf. Process. Lett. 24(2), 133–137 (1987)

    Article  MathSciNet  Google Scholar 

  19. Long Yuan, L., Qin, X.L., Chang, L., Zhang, W.: Diversified top-k clique search. VLDB J. 25(2), 171–196 (2016)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Pohang University of Science and Technology, Pohang, South Korea

    Jongmin Choi

  2. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Sergio Cabello

  3. IMFM, Ljubljana, Slovenia

    Sergio Cabello

  4. Graduate School of Artificial Intelligence, Department of Computer Science and Engineering, Pohang University of Science and Technology, Pohang, South Korea

    Hee-Kap Ahn

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  1. Jongmin Choi
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Correspondence to Hee-Kap Ahn.

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Work by Choi and Ahn was partly supported by the Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) (No. 2017-0-00905, Software Star Lab (Optimal Data Structure and Algorithmic Applications in Dynamic Geometric Environment)) and (No. 2019-0-01906, Artificial Intelligence Graduate School Program(POSTECH)). Work by Cabello was supported by the Slovenian Research Agency, program P1-0297 and projects J1-8130, J1-8155, J1-9109, J1-1693, J1-2452.

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Choi, J., Cabello, S. & Ahn, HK. Maximizing Dominance in the Plane and its Applications. Algorithmica 83, 3491–3513 (2021). https://doi.org/10.1007/s00453-021-00863-2

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  • DOI: https://doi.org/10.1007/s00453-021-00863-2

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