On the Complexity of Computing the Hypervolume Indicator 论文

摘要

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The goal of multiobjective optimization is to find a set of best compromise solutions for typically conflicting objectives. Due to the complex nature of most real-life problems, only an approximation to such an optimal set can be obtained within reasonable (computing) time. To compare such approximations, and thereby the performance of multiobjective optimizers providing them, unary quality measures are usually applied. Among these, the <emphasis emphasistype="italic">hypervolume indicator</emphasis> (or <emphasis emphasistype="italic">S-metric</emphasis>) is of particular relevance due to its favorable properties. Moreover, this indicator has been successfully integrated into stochastic optimizers, such as evolutionary algorithms, where it serves as a guidance criterion for finding good approximations to the Pareto front. Recent results show that computing the hypervolume indicator can be seen as solving a specialized version of Klee's Measure Problem. In general, Klee's Measure Problem can be solved with <formula formulatype="inline"><tex Notation="TeX">${\cal O}(n \log n + n^{d/2}\log n)$</tex></formula> comparisons for an input instance of size <formula formulatype="inline"> <tex Notation="TeX">$n$</tex></formula> in <formula formulatype="inline"><tex Notation="TeX">$d$</tex></formula> dimensions; as of this writing, it is unknown whether a lower bound higher than <formula formulatype="inline"><tex Notation="TeX">$\Omega (n \log n)$</tex></formula> can be proven. In this paper, we derive a lower bound of <formula formulatype="inline"><tex Notation="TeX">$\Omega (n\log n)$</tex></formula> for the complexity of computing the hypervolume indicator in any number of dimensions <formula formulatype="inline"> <tex Notation="TeX">$d≫1$</tex></formula> by reducing the so-called <emphasis emphasistype="smcaps">uniformgap</emphasis> problem to it. For the 3-D case, we also present a matching upper bound of <formula formulatype="inline"><tex Notation="TeX">${\cal O}(n\log n)$</tex></formula> comparisons that is obtained by extending an algorithm for finding the maxima of a point set. </para>