Computing hypervolume contributions in low dimensions: Asymptotically optimal algorithm and complexity results

Abstract

Given a finite set Y ⊂ ℝd of n mutually non-dominated vectors in d ≥ 2 dimensions, the hypervolume contribution of a point y ∈ Y is the difference between the hypervolume indicator of Y and the hypervolume indicator of Y \ {y}. In multi-objective metaheuristics, hypervolume contributions are computed in several selection and bounded-size archiving procedures. This paper presents new results on the (time) complexity of computing all hypervolume contributions. It is proved that for d = 2,3 the problem has time complexity Θ(n logn n), and, for d > 3, the time complexity is bounded below by Ω(n logn n). Moreover, complexity bounds are derived for computing a single hypervolume contribution. A dimension sweep algorithm with time complexity O(n logn) and space complexity O(n) is proposed for computing all hypervolume contributions in three dimensions. It improves the complexity of the best known algorithm for d = 3 by a factor of √n. Theoretical results are complemented by performance tests on randomly generated test-problems. © 2011 Springer-Verlag.