Document Open Access Logo

Maximum Volume Subset Selection for Anchored Boxes

Authors Karl Bringmann, Sergio Cabello, Michael T. M. Emmerich

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2017.22.pdf
  • Filesize: 0.64 MB
  • 15 pages

Document Identifiers

Author Details

Karl Bringmann
Sergio Cabello
Michael T. M. Emmerich

Cite AsGet BibTex

Karl Bringmann, Sergio Cabello, and Michael T. M. Emmerich. Maximum Volume Subset Selection for Anchored Boxes. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.SoCG.2017.22

Abstract

Let B be a set of n axis-parallel boxes in d-dimensions such that each box has a corner at the origin and the other corner in the positive quadrant, and let k be a positive integer. We study the problem of selecting k boxes in B that maximize the volume of the union of the selected boxes. The research is motivated by applications in skyline queries for databases and in multicriteria optimization, where the problem is known as the hypervolume subset selection problem. It is known that the problem can be solved in polynomial time in the plane, while the best known algorithms in any dimension d>2 enumerate all size-k subsets. We show that: * The problem is NP-hard already in 3 dimensions. * In 3 dimensions, we break the enumeration of all size-k subsets, by providing an n^O(sqrt(k)) algorithm. * For any constant dimension d, we give an efficient polynomial-time approximation scheme.
Keywords
  • geometric optimization
  • subset selection
  • hypervolume indicator
  • Klee’s 23 measure problem
  • boxes
  • NP-hardness
  • PTAS

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. A. Adamaszek and A. Wiese. Approximation schemes for maximum weight independent set of rectangles. In Proc. of the 54th IEEE Symp. on Found. of Comp. Science (FOCS), pages 400-409. IEEE, 2013. Google Scholar
  2. A. Auger, J. Bader, D. Brockhoff, and E. Zitzler. Investigating and exploiting the bias of the weighted hypervolume to articulate user preferences. In Proc. of the 11th Conf. on Genetic and Evolutionary Computation, pages 563-570. ACM, 2009. Google Scholar
  3. A. Auger, J. Bader, D. Brockhoff, and E. Zitzler. Hypervolume-based multiobjective optimization: Theoretical foundations and practical implications. Theoretical Comp. Science, 425:75-103, 2012. Google Scholar
  4. J. Bader. Hypervolume-based search for multiobjective optimization: theory and methods. PhD thesis, ETH Zurich, Zurich, Switzerland, 1993. Google Scholar
  5. F. Barahona. On the computational complexity of Ising spin glass models. J. of Physics A: Mathematical and General, 15(10):3241, 1982. Google Scholar
  6. N. Beume, C. M. Fonseca, M. López-Ibáñez, L. Paquete, and J. Vahrenhold. On the complexity of computing the hypervolume indicator. IEEE Trans. on Evolutionary Computation, 13(5):1075-1082, 2009. Google Scholar
  7. N. Beume, B. Naujoks, and M. Emmerich. SMS-EMOA: Multiobjective selection based on dominated hypervolume. European J. of Operational Research, 181(3):1653-1669, 2007. Google Scholar
  8. K. Bringmann. Bringing order to special cases of Klee’s measure problem. In Int. Symp. on Mathematical Foundations of Comp. Science, pages 207-218. Springer, 2013. Google Scholar
  9. K. Bringmann and T. Friedrich. Approximating the volume of unions and intersections of high-dimensional geometric objects. Computational Geometry, 43(6):601-610, 2010. Google Scholar
  10. K. Bringmann and T. Friedrich. An efficient algorithm for computing hypervolume contributions. Evolutionary Computation, 18(3):383-402, 2010. Google Scholar
  11. K. Bringmann and T. Friedrich. Approximating the least hypervolume contributor: NP-hard in general, but fast in practice. Theoretical Comp. Science, 425:104-116, 2012. Google Scholar
  12. K. Bringmann, T. Friedrich, and P. Klitzke. Generic postprocessing via subset selection for hypervolume and epsilon-indicator. In Int. Conf. on Parallel Problem Solving from Nature, pages 518-527. Springer, 2014. Google Scholar
  13. K. Bringmann, T. Friedrich, and P. Klitzke. Two-dimensional subset selection for hypervolume and epsilon-indicator. In Proc. of the 2014 Conf. on Genetic and Evolutionary Comput., pages 589-596. ACM, 2014. Google Scholar
  14. T. M. Chan. A (slightly) faster algorithm for Klee’s measure problem. Computational Geometry, 43(3):243-250, 2010. Google Scholar
  15. T. M. Chan. Klee’s measure problem made easy. In Proc. of the 54th IEEE Symp. on Found. of Comp. Science (FOCS), pages 410-419. IEEE, 2013. Google Scholar
  16. J. Chen, X. Huang, I. A. Kanj, and G. Xia. Linear FPT reductions and computational lower bounds. In Proc. of the 36th ACM Symp. on Theory of Computing (STOC), pages 212-221. ACM, 2004. Google Scholar
  17. M. Emmerich, A. H. Deutz, and I. Yevseyeva. A Bayesian approach to portfolio selection in multicriteria group decision making. Procedia Comp. Science, 64:993-1000, 2015. Google Scholar
  18. R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Processing Lett., 12(3):133-137, 1981. Google Scholar
  19. M. R. Garey and D. S. Johnson. The rectilinear Steiner tree problem in NP complete. SIAM J. of Applied Math., 32:826-834, 1977. Google Scholar
  20. A. P. Guerreiro, C. M. Fonseca, and L. Paquete. Greedy hypervolume subset selection in low dimensions. Evolutionary Computation, 24(3):521-544, 2016. Google Scholar
  21. D. S. Hochbaum and W. Maass. Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM, 32(1):130-136, 1985. Google Scholar
  22. H. Imai and T. Asano. Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. J. of Algorithms, 4(4):310-323, 1983. Google Scholar
  23. J. D. Knowles, D. W. Corne, and M. Fleischer. Bounded archiving using the Lebesgue measure. In Proc. of the 2003 Congress on Evolutionary Computation (CEC), volume 4, pages 2490-2497. IEEE, 2003. Google Scholar
  24. T. Kuhn, C. M. Fonseca, L. Paquete, S. Ruzika, M. M. Duarte, and J. R. Figueira. Hypervolume subset selection in two dimensions: Formulations and algorithms. Evolutionary Computation, 2015. Google Scholar
  25. G. L. Miller. Finding small simple cycle separators for 2-connected planar graphs. J. Comput. Syst. Sci., 32(3):265-279, 1986. Google Scholar
  26. J. S. B. Mitchell and M. Sharir. New results on shortest paths in three dimensions. In Proc. of the 20th ACM Symp. on Computational Geometry, pages 124-133, 2004. Google Scholar
  27. G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher. An analysis of approximations for maximizing submodular set functions - I. Mathematical Programming, 14(1):265-294, 1978. Google Scholar
  28. G. Rote, K. Buchin, K. Bringmann, S. Cabello, and M. Emmerich. Selecting k points that maximize the convex hull volume (extended abstract). In JCDCG3 2016; The 19th Japan Conf. on Discrete and Computational Geometry, Graphs, and Games, pages 58-60, 9 2016. URL: http://www.jcdcgg.u-tokai.ac.jp/JCDCG3_abstracts.pdf.
  29. J. A. Storer. On minimal-node-cost planar embeddings. Networks, 14(2):181-212, 1984. Google Scholar
  30. R. Tamassia and I. G. Tollis. Planar grid embedding in linear time. IEEE Trans. on Circuits and Systems, 36(9):1230-1234, 1989. Google Scholar
  31. T. Ulrich and L. Thiele. Bounding the effectiveness of hypervolume-based (μ+λ)-archiving algorithms. In Learning and Intelligent Optimization, pages 235-249. Springer, 2012. Google Scholar
  32. L. While, P. Hingston, L. Barone, and S. Huband. A faster algorithm for calculating hypervolume. IEEE Trans. on Evolutionary Computation, 10(1):29-38, 2006. Google Scholar
  33. J. Wu and S. Azarm. Metrics for quality assessment of a multiobjective design optimization solution set. J. of Mechanical Design, 123(1):18-25, 2001. Google Scholar
  34. E. Zitzler, L. Thiele, M. Laumanns, C. M. Fonseca, and V. G. Da Fonseca. Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. on Evolutionary Computation, 7(2):117-132, 2003. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact