Computational geometry: an introduction
Computational geometry: an introduction
Introduction to algorithms
New upper bounds in Klee's measure problem
SIAM Journal on Computing
On Finding the Maxima of a Set of Vectors
Journal of the ACM (JACM)
On the complexity of computing the measure of ∪[ai,bi]
Communications of the ACM
Multi-Objective Optimization Using Evolutionary Algorithms
Multi-Objective Optimization Using Evolutionary Algorithms
Evolutionary Algorithms for Solving Multi-Objective Problems
Evolutionary Algorithms for Solving Multi-Objective Problems
Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study
PPSN V Proceedings of the 5th International Conference on Parallel Problem Solving from Nature
The measure of Pareto optima applications to multi-objective metaheuristics
EMO'03 Proceedings of the 2nd international conference on Evolutionary multi-criterion optimization
HM'07 Proceedings of the 4th international conference on Hybrid metaheuristics
An EMO algorithm using the hypervolume measure as selection criterion
EMO'05 Proceedings of the Third international conference on Evolutionary Multi-Criterion Optimization
A new analysis of the lebmeasure algorithm for calculating hypervolume
EMO'05 Proceedings of the Third international conference on Evolutionary Multi-Criterion Optimization
Properties of an adaptive archiving algorithm for storing nondominated vectors
IEEE Transactions on Evolutionary Computation
Performance assessment of multiobjective optimizers: an analysis and review
IEEE Transactions on Evolutionary Computation
S-metric calculation by considering dominated hypervolume as klee's measure problem
Evolutionary Computation
Proceedings of the 12th annual conference on Genetic and evolutionary computation
PPSN'10 Proceedings of the 11th international conference on Parallel problem solving from nature: Part I
IEEE Transactions on Evolutionary Computation - Special issue on preference-based multiobjective evolutionary algorithms
An efficient algorithm for computing hypervolume contributions**
Evolutionary Computation
EMO'11 Proceedings of the 6th international conference on Evolutionary multi-criterion optimization
Approximating the least hypervolume contributor: NP-hard in general, but fast in practice
Theoretical Computer Science
On the properties of the R2 indicator
Proceedings of the 14th annual conference on Genetic and evolutionary computation
ICANN'12 Proceedings of the 22nd international conference on Artificial Neural Networks and Machine Learning - Volume Part II
Parameterized average-case complexity of the hypervolume indicator
Proceedings of the 15th annual conference on Genetic and evolutionary computation
General framework for localised multi-objective evolutionary algorithms
Information Sciences: an International Journal
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The goal of multiobjective optimization is to find a set of best compromise solutions for typically conflicting objectives. Due to the complex nature of most real-life problems, only an approximation to such an optimal set can be obtained within reasonable (computing) time. To compare such approximations, and thereby the performance of multiobjective optimizers providing them, unary quality measures are usually applied. Among these, the hypervolume indicator (or S-metric) is of particular relevance due to its favorable properties. Moreover, this indicator has been successfully integrated into stochastic optimizers, such as evolutionary algorithms, where it serves as a guidance criterion for finding good approximations to the Pareto front. Recent results show that computing the hypervolume indicator can be seen as solving a specialized version of Klee's Measure Problem. In general, Klee's Measure Problem can be solved with O(n log n + nd/2 log n) comparisons for an input instance of size n in d dimensions; as of this writing, it is unknown whether a lower bound higher than Ω(n log n) can be proven. In this paper, we derive a lower bound of Ω(n log n) for the complexity of computing the hypervolume indicator in any number of dimensions d 1 by reducing the so-called UNIFORMGAP problem to it. For the 3-D case, we also present a matching upper bound of O(n log n) comparisons that is obtained by extending an algorithm for finding the maxima of a point set.