Don't be greedy when calculating hypervolume contributions
Proceedings of the tenth ACM SIGEVO workshop on Foundations of genetic algorithms
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
Proceedings of the 11th Annual conference on Genetic and evolutionary computation
Articulating user preferences in many-objective problems by sampling the weighted hypervolume
Proceedings of the 11th Annual conference on Genetic and evolutionary computation
Multiplicative approximations and the hypervolume indicator
Proceedings of the 11th Annual conference on Genetic and evolutionary computation
S-metric calculation by considering dominated hypervolume as klee's measure problem
Evolutionary Computation
Approximating the volume of unions and intersections of high-dimensional geometric objects
Computational Geometry: Theory and Applications
Klee's measure problem on fat boxes in time ∂(n(d+2)/3)
Proceedings of the twenty-sixth annual symposium on Computational geometry
Integrating decision space diversity into hypervolume-based multiobjective search
Proceedings of the 12th annual conference on Genetic and evolutionary computation
Set-based multi-objective optimization, indicators, and deteriorative cycles
Proceedings of the 12th annual conference on Genetic and evolutionary computation
The maximum hypervolume set yields near-optimal approximation
Proceedings of the 12th annual conference on Genetic and evolutionary computation
Proceedings of the 12th annual conference companion 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
EMO'11 Proceedings of the 6th international conference on Evolutionary multi-criterion optimization
Proceedings of the 13th annual conference on Genetic and evolutionary computation
Achieving balance between proximity and diversity in multi-objective evolutionary algorithm
Information Sciences: an International Journal
An improved algorithm for Klee's measure problem on fat boxes
Computational Geometry: Theory and Applications
Approximating the least hypervolume contributor: NP-hard in general, but fast in practice
Theoretical Computer Science
On Klee's measure problem for grounded boxes
Proceedings of the twenty-eighth annual symposium on Computational geometry
A new multi-objective evolutionary algorithm based on a performance assessment indicator
Proceedings of the 14th annual conference on Genetic and evolutionary computation
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We consider the computation of the volume of the union ofhigh-dimensional geometric objects. While showing that this problemis #P-hard already for very simple bodies (i.e.,axis-parallel boxes), we give a fast FPRAS for all objects whereone can: (1) test whether a given point lies inside the object, (2)sample a point uniformly, (3) calculate the volume of the object inpolynomial time. All three oracles can be weak, that is, justapproximate. This implies that Klee's measure problem and thehypervolume indicator can be approximated efficiently even thoughthey are #P-hard and hence cannot be solved exactly in timepolynomial in the number of dimensions unless P = NP.Our algorithm also allows to approximate efficiently the volume ofthe union of convex bodies given by weak membership oracles.For the analogous problem of the intersection ofhigh-dimensional geometric objects we prove #P-hardness forboxes and show that there is no multiplicative polynomial-time2d1-z-approximation for certainboxes unless NP=BPP, but give a simple additivepolynomial-time ε-approximation.