New upper bounds in Klee's measure problem
SIAM Journal on Computing
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
Strong computational lower bounds via parameterized complexity
Journal of Computer and System Sciences
Covariance Matrix Adaptation for Multi-objective Optimization
Evolutionary Computation
Fixed-parameter evolutionary algorithms and the vertex cover problem
Proceedings of the 11th Annual conference on Genetic and evolutionary computation
A (slightly) faster algorithm for Klee's measure problem
Computational Geometry: Theory and Applications
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
Proceedings of the 12th annual conference on Genetic and evolutionary computation
An efficient algorithm for computing hypervolume contributions**
Evolutionary Computation
Variants of Evolutionary Algorithms for Real-World Applications
Variants of Evolutionary Algorithms for Real-World Applications
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
Multiplying matrices faster than coppersmith-winograd
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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
On Klee's measure problem for grounded boxes
Proceedings of the twenty-eighth annual symposium on Computational geometry
A Fast Way of Calculating Exact Hypervolumes
IEEE Transactions on Evolutionary Computation
Approximation quality of the hypervolume indicator
Artificial Intelligence
Parameterized Complexity
Speeding up many-objective optimization by Monte Carlo approximations
Artificial Intelligence
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The hypervolume indicator (HYP) is a popular measure for the quality of a set of n solutions in ℜRd. We discuss its asymptotic worst-case runtimes and several lower bounds depending on different complexity-theoretic assumptions. Assuming that P ≠ NP, there is no algorithm with runtime poly(n,d). Assuming the exponential time hypothesis, there is no algorithm with runtime no(d). In contrast to these worst-case lower bounds, we study the average-case complexity of HYP for points distributed i.i.d. at random on a d-dimensional simplex. We present a general framework which translates any algorithm for HYP with worst-case runtime n f(d) to an algorithm with worst-case runtime n f(d)+1 and fixed-parameter-tractable (FPT) average-case runtime. This can be used to show that HYP can be solved in expected time O(d d2/2, n + d, n2), which implies that HYP is FPT on average while it is W[1]-hard in the worst-case. For constant dimension d this gives an algorithm for HYP with runtime O(n2) on average. This is the first result proving that HYP is asymptotically easier in the average case. It gives a theoretical explanation why most HYP algorithms perform much better on average than their theoretical worst-case runtime predicts.