Approximating the least hypervolume contributor: NP-hard in general, but fast in practice
- 作者：Karl Bringmanna ; Tobias Friedrichb ;
- 关键词：Evolutionary algorithms ; Multi-objective optimization ; Hypervolume indicator
- 刊名：Theoretical Computer Science
- 出版年：2012
- 期刊代码：114_03043975
- 类别：cp
- 出版时间：30 March, 2012
- 卷：425
- 期：Complete
- 页码：104-116
- 文件大小：830 K
摘要
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 times the minimal contribution is NP-hard. This implies that it is neither possible to efficiently find the least contributing solution (unless ) nor to approximate it (unless ).
Nevertheless, in the second part of the paper we present a fast approximation algorithm for this problem. We prove that for arbitrarily given it calculates a solution with contribution at most times the minimal contribution with probability at least . 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.