Abstract
Given a finite set Y ⊂ ℝd of n mutually non-dominated vectors in d ≥ 2 dimensions, the hypervolume contribution of a point y ∈ Y is the difference between the hypervolume indicator of Y and the hypervolume indicator of Y ∖ {y}. In multi-objective metaheuristics, hypervolume contributions are computed in several selection and bounded-size archiving procedures.
This paper presents new results on the (time) complexity of computing all hypervolume contributions. It is proved that for d = 2,3 the problem has time complexity Θ(n logn), and, for d > 3, the time complexity is bounded below by Ω(n logn). Moreover, complexity bounds are derived for computing a single hypervolume contribution.
A dimension sweep algorithm with time complexity \(\mathcal{O}\)(n logn) and space complexity \(\mathcal{O}(n)\) is proposed for computing all hypervolume contributions in three dimensions. It improves the complexity of the best known algorithm for d = 3 by a factor of \(\sqrt{n}\). Theoretical results are complemented by performance tests on randomly generated test-problems.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Adelson-Velskij, G., Landis, E.: An algorithm for the organization of information. Doklady Akad. Nauk SSSR 146, 263–266 (1962)
Bader, J., Zitzler, E.: HypE: An algorithm for fast hypervolume-based many-objective optimization. Evolutionary Computation (2010) (in press)
Beume, N.: S-metric calculation by considering dominated hypervolume as Klee’s measure problem. Evolutionary Computation 17(4), 477–492 (2009)
Beume, N., Fonseca, C.M., López-Ibáñez, M., Paquete, L., Vahrenhold, J.: On the complexity of computing the hypervolume indicator. IEEE Transact. Evolutionary Computation 13(5), 1075–1082 (2009)
Bradstreet, L., Barone, L., While, L.: Updating exclusive hypervolume contributions cheaply. In: Conf. on Evolutionary Computation, pp. 538–544. IEEE Press, Los Alamitos (2009)
Bradstreet, L., While, L., Barone, L.: A fast incremental hypervolume algorithm. IEEE Transact. on Evolutionary Computation 12(6), 714–723 (2008)
Bringmann, K., Friedrich, T.: Approximating the volume of unions and intersections of high-dimensional geometric objects. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 436–447. Springer, Heidelberg (2008)
Bringmann, K., Friedrich, T.: Approximating the least hypervolume contributor: NP-hard in general, but fast in practice. In: Ehrgott, M., Fonseca, C.M., Gandibleux, X., Hao, J.-K., Sevaux, M. (eds.) EMO 2009. LNCS, vol. 5467, pp. 6–20. Springer, Heidelberg (2009)
Bringmann, K., Friedrich, T.: An efficient algorithm for computing hypervolume contributions. Evolutionary Computation 18(3), 383–402 (2010)
Emmerich, M., Beume, N., Naujoks, B.: An EMO algorithm using the hypervolume measure as selection criterion. In: Coello Coello, C.A., Hernández Aguirre, A., Zitzler, E. (eds.) EMO 2005. LNCS, vol. 3410, pp. 62–76. Springer, Heidelberg (2005)
Fleischer, M.: The measure of pareto optima applications to multi-objective metaheuristics. In: Fonseca, C.M., Fleming, P.J., Zitzler, E., Deb, K., Thiele, L. (eds.) EMO 2003. LNCS, vol. 2632, pp. 519–533. Springer, Heidelberg (2003)
Fonseca, C.M., Paquete, L., López-Ibáñez, M.: An improved dimension-sweep algorithm for the hypervolume indicator. In: Conf. on Evolutionary Computation, pp. 1157–1163. IEEE Press, Los Alamitos (2006)
Huband, S., Hingston, P., While, L., Barone, L.: An evolution strategy with probabilistic mutation for multi-objective optimisation. In: Conf. on Evolutionary Computation, vol. 4, pp. 2284–2291. IEEE Press, Los Alamitos (2003)
Igel, C., Hansen, N., Roth, S.: Covariance matrix adaptation for multi-objective optimization. Evolutionary Computation 15(1), 1–28 (2007)
Ishibuchi, H., Tsukamoto, N., Sakane, Y., Nojima, Y.: Indicator-based evolutionary algorithm with hypervolume approximation by achievement scalarizing functions. In: GECCO 2010, pp. 527–534. ACM, USA (2010)
Knowles, J., Corne, D., Fleischer, M.: Bounded archiving using the Lebesgue measure. In: Conf. on Evolutionary Computation, pp. 2490–2497. IEEE Press, Los Alamitos (2003)
Knuth, D.: The Art of Computer Programming, vol. 3. Addison-Wesley, Reading (1998)
Kung, H.T., Luccio, F., Preparata, F.P.: On finding the maxima of a set of vectors. J. ACM 22(4), 469–476 (1975)
Mostaghim, S., Branke, J., Schmeck, H.: Multi-objective particle swarm optimization on computer grids. In: GECCO 2007, pp. 869–875. ACM, New York (2007)
Naujoks, B., Beume, N., Emmerich, M.: Multi-objective optimisation using S-metric selection: Application to three-dimensional solution spaces, vol. 2 (2005)
Overmars, M.H., Yap, C.K.: New upper bounds in klee’s measure problem. SIAM J. Comput. 20(6), 1034–1045 (1991)
Preparata, F.P., Shamos, M.I.: Computational Geometry. Springer, Heidelberg (1985)
While, L., Hingston, P., Barone, L., Huband, S.: A faster algorithm for calculating hypervolume. IEEE Transact. Evolutionary Computation 10(1), 29–38 (2006)
Zitzler, E.: Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications. Ph.D. thesis, ETH Zurich, Switzerland (1999)
Zitzler, E., Künzli, S.: Indicator-based selection in multiobjective search. In: Yao, X., Burke, E.K., Lozano, J.A., Smith, J., Merelo-Guervós, J.J., Bullinaria, J.A., Rowe, J.E., Tiňo, P., Kabán, A., Schwefel, H.-P. (eds.) PPSN 2004. LNCS, vol. 3242, pp. 832–842. Springer, Heidelberg (2004)
Zitzler, E., Thiele, L.: Multiobjective optimization using evolutionary algorithms - A comparative case study. In: Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.-P. (eds.) PPSN 1998. LNCS, vol. 1498, pp. 292–301. Springer, Heidelberg (1998)
Zitzler, E., Thiele, L., Laumanns, M., Foneseca, C.M., Grunert da Fonseca, V.: Performance assessment of multiobjective optimizers: An analysis and review. IEEE Transact. on Evolutionary Computation 7(2), 117–132 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Emmerich, M.T.M., Fonseca, C.M. (2011). Computing Hypervolume Contributions in Low Dimensions: Asymptotically Optimal Algorithm and Complexity Results. In: Takahashi, R.H.C., Deb, K., Wanner, E.F., Greco, S. (eds) Evolutionary Multi-Criterion Optimization. EMO 2011. Lecture Notes in Computer Science, vol 6576. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19893-9_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-19893-9_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-19892-2
Online ISBN: 978-3-642-19893-9
eBook Packages: Computer ScienceComputer Science (R0)