Computing and Updating Hypervolume Contributions in Up to Four Dimensions

Arising in connection with multiobjective selection and archiving, the hypervolume subset selection problem (HSSP) consists in finding a subset of size ${k\leq n}$ of a set ${X\subset \mathbb {R}^{d}}$ of ${n}$ nondominated points in ${d}$ -dimensional space that maximizes the hypervolume indicator. The incremental greedy approximation to the HSSP has an approximation guarantee of ${1-1/e}$ , and is polynomial in ${n}$ and ${k}$ , while no polynomial exact algorithms are known for ${d\geq 3}$ . The decremental greedy counterpart has no known approximation guarantee, but is potentially faster for large ${k}$ , and still leads to good approximations in practice. The computation and update of individual hypervolume contributions are at the core of the implementation of this greedy strategy. In this paper, new algorithms for the computation and update of hypervolume contributions are developed. In three dimensions, updating the total hypervolume and all individual contributions under single-point changes is performed in linear time, while in the 4-D case all contributions are computed in ${O(n^{2})}$ time. As a consequence, the decremental greedy approximation to the HSSP can now be obtained in ${O(n(n-k) + n\log n)}$ and ${O(n^{2}(n-k))}$ time for ${d=3}$ and ${d=4}$ , respectively. Experimental results show that the proposed algorithms significantly outperform existing ones.