Approximating the Least Hypervolume Contributor: NP-Hard in General, But Fast in Practice

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+ε) 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 very fast approximation algorithm for this problem. We prove that for arbitrarily given ε, δ > 0 it calculates a solution with contribution at most (1 + ε) times the minimal contribution with probability at least (1−δ). 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.

Bringing Order to Special Cases of Klee’s Measure Problem

“…Improving upon the general case of KMP, there is an algorithm with runtime O(n log n) for d = 3 [4]. The same paper also shows an unconditional lower bound of Ω(n log n) for d > 1, while #P -hardness in the number of dimensions was shown in [8]. Recently, an algorithm with runtime O(n (d−1)/2 log n) for d 3 was presented in [19].…”

R2-Based Hypervolume Contribution Approximation

“…In this situation, the line segments in the hypervolume contribution region of the solution s not only intersect with the attainment surface of the solution set A \ {s} but also intersect with the hypervolume boundary of the solution set A associated with the reference point r. For the line segments intersecting with the attainment surface of the solution set A \ {s}, the lengths can be calculated by Eq. (9). For the line segments intersecting with the hypervolume boundary associated with the reference point r, the lengths are calculated as follows:…”

“…Whereas several fast hypervolume calculation methods [2], [3], [4], [5] have been proposed, it has been proved that the exact hypervolume calculation is #P-hard in the number of dimensions [6]. Therefore, efforts in the hypervolume approximation have been done to increase the applicability of the hypervolume to high-dimensional spaces, including the Monte Carlo sampling method [7], [8], [9] and the achievement scalarizing function method [10], [11].…”

Tutorial on Evolutionary Multiobjective Optimization

“…SOM for supersonic wing design [Obayashi and Sasaki 2003] Biclustering for processor design and knapsack [Ulrich et al 2007] Successful case studies in engineering (noise barrier design, polymer extrusion, friction stir welding) [Deb et al 2014] But: can also result in cycles if reference point is not constant [Judt et al 2011] and is expensive to compute exactly [Bringmann and Friedrich 2009] …”