The hypervolume subset selection problem (HSSP) aims at approximating a set of n multidimensional points in Rd with an optimal subset of a given size. The size k of the subset is a parameter of the problem, and an approximation is considered best when it maximizes the hypervolume indicator. This problem has proved popular in recent years as a procedure for multiobjective evolutionary algorithms. Efficient algorithms are known for planar points (d=2), but there are hardly any results on HSSP in larger dimensions (d≥3). So far, most algorithms in higher dimensions essentially enumerate all possible subsets to determine the optimal one, and most of the effort has been directed toward improving the efficiency of hypervolume computation. We propose efficient algorithms for the selection problem in dimension 3 when either k or n-k is small, and extend our techniques to arbitrary dimensions for k≤3.

中文翻译：

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具有小子集的超体积子集选择

超体积子集选择问题 (HSSP) 旨在用给定大小的最佳子集逼近 Rd 中的一组 n 多维点。子集的大小 k 是问题的一个参数，当它最大化 hypervolume 指标时，近似值被认为是最好的。近年来，这个问题已被证明是多目标进化算法的一个程序。对于平面点（d=2），已知有效的算法，但在更大维度（d≥3）的 HSSP 上几乎没有任何结果。到目前为止，大多数更高维度的算法本质上是枚举所有可能的子集来确定最佳子集，并且大部分工作都集中在提高超体积计算的效率上。当 k 或 nk 较小时，我们为维度 3 中的选择问题提出了有效的算法，