A Fast Dimension-Sweep Algorithm for the Hypervolume Indicator in Four Dimensions

The Hypervolume Indicator is one of the most widely used quality indicators in Evolutionary Multiobjective Optimization. Its computation is a special case of Klee’s Measure Problem (KMP) where the upper end of all rectangular ranges coincides with a given reference point (assuming minimization, without loss of generality). Although the time complexity of the hypervolume indicator in two and three dimensions is known to be Θ(n log n), improving upon the O(n log n) complexity of Overmars and Yap’s algorithm for the general KMP in higher dimensions has been a challenge. In this paper, a new dimension-sweep algorithm to compute the hypervolume indicator in four dimensions is proposed, and its complexity is shown to be O(n).