Computing efficiently the nondominated subset of a set sum

We study the computation of the nondominated subset of the Minkowski sum (or set sum) of a finite family of finite sets of multidimensional, real-valued points. When points are the images of solutions to multiobjective optimization problems, this computation is a key algorithmic component for solving coupled systems problems. At first sight, this computation may be thought of as computing all combinations of points and applying dominance filtering to the resulting set of points. However, structural properties can be exploited algorithmically to perform this task more efficiently, avoiding some comparisons. We propose two main approaches of doing so. First, we show that using lexicographic ordering over each set to be added to the sum, we can speed up the sorting phase required to efficiently apply dominance filtering to elements of the set sum. Second, we introduce box-based methods, defining dominance relations between sets of combinations of points.

Automated summary

This paper studies the algorithmic problem of computing the nondominated subset of the Minkowski sum of finite sets of multidimensional, real-valued points. By exploiting structural properties and introducing two main approaches, the computation can be performed more efficiently, avoiding unnecessary comparisons.