Approximating the least hypervolume contributor: NP-hard in general, but fast in practice

TL;DR

This paper proves that calculating the exact hypervolume contribution is NP-hard, but introduces a fast probabilistic approximation algorithm that performs well on large datasets, enabling practical use in high-dimensional optimization.

Contribution

It establishes the NP-hardness of exact hypervolume contribution calculation and presents a fast approximation algorithm with probabilistic guarantees.

Findings

Exact calculation is #P-hard and NP-hard to approximate.

The proposed algorithm efficiently handles large, high-dimensional datasets.

The algorithm achieves near-optimal solutions within seconds on benchmark datasets.

Abstract

The hypervolume indicator is an increasingly popular set measure to compare the quality of two Pareto sets. The basic ingredient of most hypervolume indicator based optimization algorithms is the calculation of the hypervolume contribution of single solutions regarding a Pareto set. We show that exact calculation of the hypervolume contribution is #P-hard while its approximation is NP-hard. The same holds for the calculation of the minimal contribution. We also prove that it is NP-hard to decide whether a solution has the least hypervolume contribution. Even deciding whether the contribution of a solution is at most $(1+\eps)$ times the minimal contribution is NP-hard. This implies that it is neither possible to efficiently find the least contributing solution (unless $P = NP$) nor to approximate it (unless $NP = BPP$). Nevertheless, in the second part of the paper we present a fast approximation algorithm for this problem. We prove that for arbitrarily given $\eps,\delta>0$ it calculates a solution with contribution at most $(1+\eps)$ times the minimal contribution with probability at least $(1-\delta)$. Though it cannot run in polynomial time for all instances, it performs extremely fast on various benchmark datasets. The algorithm solves very large problem instances which are intractable for exact algorithms (e.g., 10000 solutions in 100 dimensions) within a few seconds.