New upper bounds in Klee's measure problem
SIAM Journal on Computing
On the hardness of approximate reasoning
Artificial Intelligence
A note on a method for generating points uniformly on n-dimensional spheres
Communications of the ACM
A Fast Elitist Non-dominated Sorting Genetic Algorithm for Multi-objective Optimisation: NSGA-II
PPSN VI Proceedings of the 6th International Conference on Parallel Problem Solving from Nature
A Fast Algorithm for Computing the Contribution of a Point to the Hypervolume
ICNC '07 Proceedings of the Third International Conference on Natural Computation - Volume 04
Approximating the Volume of Unions and Intersections of High-Dimensional Geometric Objects
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
Don't be greedy when calculating hypervolume contributions
Proceedings of the tenth ACM SIGEVO workshop on Foundations of genetic algorithms
S-metric calculation by considering dominated hypervolume as klee's measure problem
Evolutionary Computation
On the complexity of computing the hypervolume indicator
IEEE Transactions on Evolutionary Computation
Approximating the volume of unions and intersections of high-dimensional geometric objects
Computational Geometry: Theory and Applications
Bandit-based estimation of distribution algorithms for noisy optimization: rigorous runtime analysis
LION'10 Proceedings of the 4th international conference on Learning and intelligent optimization
Hype: An algorithm for fast hypervolume-based many-objective optimization
Evolutionary Computation
An EMO algorithm using the hypervolume measure as selection criterion
EMO'05 Proceedings of the Third international conference on Evolutionary Multi-Criterion Optimization
Multiobjective evolutionary algorithms: a comparative case studyand the strength Pareto approach
IEEE Transactions on Evolutionary Computation
Performance assessment of multiobjective optimizers: an analysis and review
IEEE Transactions on Evolutionary Computation
A faster algorithm for calculating hypervolume
IEEE Transactions on Evolutionary Computation
A Fast Incremental Hypervolume Algorithm
IEEE Transactions on Evolutionary Computation
Convergence of hypervolume-based archiving algorithms ii: competitiveness
Proceedings of the 14th annual conference on Genetic and evolutionary computation
Approximation quality of the hypervolume indicator
Artificial Intelligence
Parameterized average-case complexity of the hypervolume indicator
Proceedings of the 15th annual conference on Genetic and evolutionary computation
A note on the ε-indicator subset selection
Theoretical Computer Science
Speeding up many-objective optimization by Monte Carlo approximations
Artificial Intelligence
Hi-index | 5.23 |
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+@e) 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 fast approximation algorithm for this problem. We prove that for arbitrarily given @e,@d0 it calculates a solution with contribution at most (1+@e) times the minimal contribution with probability at least (1-@d). 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., 10,000 solutions in 100 dimensions) within a few seconds.