# Greedy Hypervolume Subset Selection in the Three-Objective Case

## Pedro Matias

Abstract: Given a non-dominated point set $X \subset \mathbb{R}^d$ of size $n$ and a suitable reference point $r \in \mathbb{R}^d$, the Hypervolume Subset Selection Problem (HSSP) consists of finding a subset of size $k \le n$ that maximizes the hypervolume indicator. It arises in connection with multiobjective selection and archiving strategies, as well as Pareto-front approximation post-processing 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 $O(n^{\frac{d}{2}} \log n + n^{n-k} )$. 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. Such a greedy algorithm for the 3-dimensional HSSP is proposed in this paper. The time complexity of the algorithm is shown to be $O(n(k + \log n))$, which considerably improves upon recent complexity results for this approximation problem.

Paper by Guerreiro, Fonseca, and Paquete, at GECCO '15, Annual Conference on Genetic and Evolutionary Computation