Computational geometry: an introduction
Computational geometry: an introduction
New upper bounds in Klee's measure problem
SIAM Journal on Computing
The art of computer programming, volume 3: (2nd ed.) sorting and searching
The art of computer programming, volume 3: (2nd ed.) sorting and searching
On Finding the Maxima of a Set of Vectors
Journal of the ACM (JACM)
Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study
PPSN V Proceedings of the 5th International Conference on Parallel Problem Solving from Nature
Covariance Matrix Adaptation for Multi-objective Optimization
Evolutionary Computation
Multi-objective particle swarm optimization on computer grids
Proceedings of the 9th annual conference on Genetic and evolutionary computation
Approximating the Volume of Unions and Intersections of High-Dimensional Geometric Objects
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
Approximating the Least Hypervolume Contributor: NP-Hard in General, But Fast in Practice
EMO '09 Proceedings of the 5th International Conference on Evolutionary Multi-Criterion Optimization
S-metric calculation by considering dominated hypervolume as klee's measure problem
Evolutionary Computation
Updating exclusive hypervolume contributions cheaply
CEC'09 Proceedings of the Eleventh conference on Congress on Evolutionary Computation
On the complexity of computing the hypervolume indicator
IEEE Transactions on Evolutionary Computation
The measure of Pareto optima applications to multi-objective metaheuristics
EMO'03 Proceedings of the 2nd international conference on Evolutionary multi-criterion optimization
Proceedings of the 12th annual conference on Genetic and evolutionary computation
An efficient algorithm for computing hypervolume contributions**
Evolutionary Computation
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
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
Speeding up many-objective optimization by Monte Carlo approximations
Artificial Intelligence
| Hi-index | 0.00 |
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 log n), and, for d 3, the time complexity is bounded below by Ω(n log n). Moreover, complexity bounds are derived for computing a single hypervolume contribution. A dimension sweep algorithm with time complexity O(n log n) and space complexity 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 √n. Theoretical results are complemented by performance tests on randomly generated test-problems.