On the Complexity of Computing the Hypervolume Indicator

The goal of multiobjective optimization is to find a set of best compromise solutions for typically conflicting objectives. Due to the complex nature of most real-life problems, only an approximation to such an optimal set can be obtained within reasonable (computing) time. To compare such approximations, and thereby the performance of multiobjective optimizers providing them, unary quality measures are usually applied. Among these, the hypervolume indicator (or S-metric) is of particular relevance due to its favorable properties. Moreover, this indicator has been successfully integrated into stochastic optimizers, such as evolutionary algorithms, where it serves as a guidance criterion for finding good approximations to the Pareto front. Recent results show that computing the hypervolume indicator can be seen as solving a specialized version of Klee's Measure Problem. In general, Klee's Measure Problem can be solved with O(n log n + n(d/2) log n) comparisons for an input instance of size n in d dimensions; as of this writing, it is unknown whether a lower bound higher than Omega(n log n) can be proven. In this paper, we derive a lower bound of Omega(n log n) for the complexity of computing the hypervolume indicator in any number of dimensions d > 1 by reducing the so-called UNIFORMGAP problem to it. For the 3-D case, we also present a matching upper bound of O(n log n) comparisons that is obtained by extending an algorithm for finding the maxima of a point set.