Abstract
The hypervolume indicator is an increasingly popular set measure to compare the quality of two Pareto sets. The basic ingredient of most hypervolume indicator based optimization algorithms is the calculation of the hypervolume contribution of single solutions regarding a Pareto set. We show that exact calculation of the hypervolume contribution is #P-hard while its approximation is NP-hard. The same holds for the calculation of the minimal contribution. We also prove that it is NP-hard to decide whether a solution has the least hypervolume contribution. Even deciding whether the contribution of a solution is at most (1 + ε) times the minimal contribution is NP-hard. This implies that it is neither possible to efficiently find the least contributing solution (unless P = NP) nor to approximate it (unless NP = BPP).
Nevertheless, in the second part of the paper we present a very fast approximation algorithm for this problem. We prove that for arbitrarily given ε,δ> 0 it calculates a solution with contribution at most (1 + ε) times the minimal contribution with probability at least (1 − δ). Though it cannot run in polynomial time for all instances, it performs extremely fast on various benchmark datasets. The algorithm solves very large problem instances which are intractable for exact algorithms (e.g., 10000 solutions in 100 dimensions) within a few seconds.
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References
Bader, J., Zitzler, E.: HypE: An Algorithm for Fast Hypervolume-Based Many-Objective Optimization. TIK Report 286, Institut für Technische Informatik und Kommunikationsnetze, ETH Zürich (2008)
Beume, N., Rudolph, G.: Faster S-metric calculation by considering dominated hypervolume as Klee’s measure problem. In: Proc. Second International Conference on Computational Intelligence (IASTED 2006), pp. 233–238 (2006)
Beume, N., Fonseca, C., López-Ibáñez, M., Paquete, L., Vahrenhold, J.: On the complexity of computing the hypervolume indicator. Technical report CI-235/07, Technical University of Dortmund (2007)
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 (2008a), http://arxiv.org/abs/0809.0835
Bringmann, K., Friedrich, T.: Approximating the least hypervolume contributor: NP-hard in general, but fast in practice (2008a), http://arxiv.org/abs/0812.2636
Deb, K., Agrawal, S., Pratap, A., Meyarivan, T.: A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II. In: Deb, K., Rudolph, G., Lutton, E., Merelo, J.J., Schoenauer, M., Schwefel, H.-P., Yao, X. (eds.) PPSN 2000. LNCS, vol. 1917, pp. 849–858. Springer, Heidelberg (2000)
Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable multi-objective optimization test problems. In: Proc. IEEE Congress on Evolutionary Computation (CEC 2002), pp. 825–830 (2002)
Klee, V.: Can the measure of ∪ [a i , b i ] be computed in less than O(n logn) steps? American Mathematical Monthly 84, 284–285 (1977)
Knowles, J.D.: Local-Search and Hybrid Evolutionary Algorithms for Pareto Optimization. PhD thesis, Department of Computer Science, University of Reading, UK (2002)
Muller, M.E.: A note on a method for generating points uniformly on n-dimensional spheres. Commun. ACM 2, 19–20 (1959)
Overmars, M.H., Yap, C.-K.: New upper bounds in Klee’s measure problem. SIAM J. Comput. 20, 1034–1045 (1991); announced at 29th Annual Symposium on Foundations of Computer Science (FOCS 1988)
Papadimitriou, C.M.: Computational Complexity. Addison-Wesley Publishing Company, Reading (1994)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes: The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1992)
Roth, D.: On the hardness of approximate reasoning. Artif. Intell. 82, 273–302 (1996)
Valiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979)
While, R.L., Bradstreet, L., Barone, L., Hingston, P.: Heuristics for optimizing the calculation of hypervolume for multi-objective optimization problems. In: Proc. IEEE Congress on Evolutionary Computation (CEC 2005), pp. 2225–2232 (2005)
While, R.L., Hingston, P., Barone, L., Huband, S.: A faster algorithm for calculating hypervolume. IEEE Trans. Evolutionary Computation 10, 29–38 (2006)
Zhou, X., Mao, N., Li, W., Sun, C.: A fast algorithm for computing the contribution of a point to the hypervolume. In: Proc. Third International Conference on Natural Computation (ICNC 2007), vol. 4, pp. 415–420 (2007)
Zitzler, E.: Hypervolume metric calculation, 2001. Computer Engineering and Networks Laboratory (TIK), ETH Zurich, Switzerland, ftp://ftp.tik.ee.ethz.ch/pub/people/zitzler/hypervol.c
Zitzler, E., Thiele, L.: Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans. Evolutionary Computation 3, 257–271 (1999); announced at 5th International Conference Parallel Problem Solving from Nature (PPSN V)
Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the strength pareto evolutionary algorithm for multiobjective optimization. In: Giannakoglou, K., et al. (eds.) Proc. Evolutionary Methods for Design, Optimisation and Control with Application to Industrial Problems (EUROGEN 2001), pp. 95–100. International Center for Numerical Methods in Engineering (CIMNE) (2002)
Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., da Fonseca, V.G.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evolutionary Computation 7, 117–132 (2003)
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Bringmann, K., Friedrich, T. (2009). Approximating the Least Hypervolume Contributor: NP-Hard in General, But Fast in Practice. In: Ehrgott, M., Fonseca, C.M., Gandibleux, X., Hao, JK., Sevaux, M. (eds) Evolutionary Multi-Criterion Optimization. EMO 2009. Lecture Notes in Computer Science, vol 5467. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01020-0_6
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DOI: https://doi.org/10.1007/978-3-642-01020-0_6
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