Let B be a set of n axis-parallel boxes in d-dimensions such that each box has a corner at the origin and the other corner in the positive quadrant, and let k be a positive integer. We study the problem of selecting k boxes in B that maximize the volume of the union of the selected boxes. The research is motivated by applications in skyline queries for databases and in multicriteria optimization, where the problem is known as the hypervolume subset selection problem. It is known that the problem can be solved in polynomial time in the plane, while the best known algorithms in any dimension d>2 enumerate all size-k subsets. We show that: * The problem is NP-hard already in 3 dimensions. * In 3 dimensions, we break the enumeration of all size-k subsets, by providing an n^O(sqrt(k)) algorithm. * For any constant dimension d, we give an efficient polynomial-time approximation scheme.
@cite_35: 
@cite_21: The hypervolume subset selection problem consists of finding a subset, with a given cardinality k, of a set of nondominated points that maximizes the hypervolume indicator. This problem arises in selection procedures of evolutionary algorithms for multiobjective optimization, for which practically efficient algorithms are required. In this article, two new formulations are provided for the two-dimensional variant of this problem. The first is a linear integer programming formulation that can be solved by solving its linear programming relaxation. The second formulation is a k-link shortest path formulation on a special digraph with the Monge property that can be solved by dynamic programming in time. This improves upon the result of in Bader 2009, and slightly improves upon the result of in Bringmann eti¾ al. 2014b, which was developed independently from this work using different techniques. Numerical results are shown for several values of n and k.
@cite_22: The goal of bi-objective optimization is to find a small set of good compromise solutions. A common problem for bi-objective evolutionary algorithms is the following subset selection problem (SSP): Given n solutions P ⊂ R2 in the objective space, select k solutions P* from P that optimize an indicator function. In the hypervolume SSP we want to select k points P* that maximize the hypervolume indicator IHYP(P*, r) for some reference point r ∈ R2. Similarly, the e-indicator SSP aims at selecting k points P* that minimize the e-indicator Ie(P*,R) for some reference set R ⊂ R2 of size m (which can be R=P). We first present a new algorithm for the hypervolume SSP with runtime O(n (k + log n)). Our second main result is a new algorithm for the e-indicator SSP with runtime O(n log n + m log m). Both results improve the current state of the art runtimes by a factor of (nearly) @math and make the problems tractable for new applications. Preliminary experiments confirm that the theoretical results translate into substantial empirical runtime improvements.
@cite_6: Optimizing the hypervolume indicator within evolutionary multiobjective optimizers has become popular in the last years. Recently, the indicator has been generalized to the weighted case to incorporate various user preferences into hypervolume-based search algorithms. There are two main open questions in this context: (i) how does the specified weight influence the distribution of a fixed number of points that maximize the weighted hypervolume indicator? (ii) how can the user articulate her preferences easily without specifying a certain weight distribution function? In this paper, we tackle both questions. First, we theoretically investigate optimal distributions of μ points that maximize the weighted hypervolume indicator. Second, based on the obtained theoretical results, we propose a new approach to articulate user preferences within biobjective hypervolume-based optimization in terms of specifying a desired density of points on a predefined (imaginary) Pareto front. Within this approach, a new exact algorithm based on dynamic programming is proposed which selects the set of μ points that maximizes the (weighted) hypervolume indicator. Experiments on various test functions show the usefulness of this new preference articulation approach and the agreement between theory and practice.
* Known Results for Volume Selection As mentioned above, 2-dimensional can be solved in polynomial time; the initial @math algorithm @cite_6 was later improved to @math @cite_22 @cite_21 . In higher dimensions, by enumerating all size- @math subsets and solving an instance of Klee's measure problem on anchored boxes for each one, there is an @math algorithm. For small @math , this can be improved to @math @cite_35 . is NP-hard when @math is part of the input, since the same holds already for Klee's measure problem on anchored boxes. However, this does not explain the exponential dependence on @math for constant @math .