Abstract
Given a nondominated point set of size
and a suitable reference point
, the Hypervolume Subset Selection Problem (HSSP) consists of finding a subset of size
that maximizes the hypervolume indicator. It arises in connection with multiobjective selection and archiving strategies, as well as Pareto-front approximation postprocessing for visualization and/or interaction with a decision maker. Efficient algorithms to solve the HSSP are available only for the 2-dimensional case, achieving a time complexity of
. In contrast, the best upper bound available for
is
. Since the hypervolume indicator is a monotone submodular function, the HSSP can be approximated to a factor of
using a greedy strategy. In this article, greedy
-time algorithms for the HSSP in 2 and 3 dimensions are proposed, matching the complexity of current exact algorithms for the 2-dimensional case, and considerably improving upon recent complexity results for this approximation problem.