Hypervolume Subset Selection with Small Subsets

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 $k≤3$⁠.