A Box Decomposition Algorithm to Compute the Hypervolume Indicator

TL;DR

This paper introduces a novel box decomposition approach for computing the hypervolume indicator, offering algorithms that are practically efficient and competitive with existing methods, with proven bounds on complexity.

Contribution

A new nonincremental and incremental box decomposition algorithm for hypervolume computation, with improved theoretical complexity bounds and practical efficiency.

Findings

Algorithms are practically efficient and competitive.

Proven upper and lower bounds on worst-case complexity.

Enhanced theoretical bounds for existing hypervolume algorithms.

Abstract

We propose a new approach to the computation of the hypervolume indicator, based on partitioning the dominated region into a set of axis-parallel hyperrectangles or boxes. We present a nonincremental algorithm and an incremental algorithm, which allows insertions of points, whose time complexities are $O(n^{\lfloor \frac{p-1}{2} \rfloor+1})$ and $O(n^{\lfloor \frac{p}{2} \rfloor+1})$, respectively. While the theoretical complexity of such a method is lower bounded by the complexity of the partition, which is, in the worst-case, larger than the best upper bound on the complexity of the hypervolume computation, we show that it is practically efficient. In particular, the nonincremental algorithm competes with the currently most practically efficient algorithms. Finally, we prove an enhanced upper bound of $O(n^{p-1})$ and a lower bound of $\Omega (n^{\lfloor \frac{p}{2}\rfloor} \log n )$ for $p \geq 4$ on the worst-case complexity of the WFG algorithm.