Greedy Hypervolume Subset Selection in Low Dimensions

Given a nondominated point set X subset of R-d of size n and a suitable reference point r is an element of R-d, the Hypervolume Subset Selection Problem (HSSP) consists of finding a subset of size k <= n 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 O(n(k + log n)). In contrast, the best upper bound available for d > 2 is O(n(d/2) log n + n(n-k)). Since the hypervolume indicator is a monotone submodular function, the HSSP can be approximated to a factor of (1 - 1/e) using a greedy strategy. In this article, greedy O(n(k + log n))-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.