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

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.Comment: 22 pages, to appear in Theoretical Computer Scienc

Stability of Curvature Measures

We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive $\mu$-reach can be estimated by the same curvature measures of the offset of a compact set K' close to K in the Hausdorff sense. We show how these curvature measures can be computed for finite unions of balls. The curvature measures of the offset of a compact set with positive $\mu$-reach can thus be approximated by the curvature measures of the offset of a point-cloud sample. These results can also be interpreted as a framework for an effective and robust notion of curvature

Worst-Case Efficient Dynamic Geometric Independent Set

We consider the problem of maintaining an approximate maximum independent set of geometric objects under insertions and deletions. We present a data structure that maintains a constant-factor approximate maximum independent set for broad classes of fat objects in d dimensions, where d is assumed to be a constant, in sublinear worst-case update time. This gives the first results for dynamic independent set in a wide variety of geometric settings, such as disks, fat polygons, and their high-dimensional equivalents. For axis-aligned squares and hypercubes, our result improves upon all (recently announced) previous works. We obtain, in particular, a dynamic (4+?)-approximation for squares, with O(log? n) worst-case update time. Our result is obtained via a two-level approach. First, we develop a dynamic data structure which stores all objects and provides an approximate independent set when queried, with output-sensitive running time. We show that via standard methods such a structure can be used to obtain a dynamic algorithm with amortized update time bounds. Then, to obtain worst-case update time algorithms, we develop a generic deamortization scheme that with each insertion/deletion keeps (i) the update time bounded and (ii) the number of changes in the independent set constant. We show that such a scheme is applicable to fat objects by showing an appropriate generalization of a separator theorem. Interestingly, we show that our deamortization scheme is also necessary in order to obtain worst-case update bounds: If for a class of objects our scheme is not applicable, then no constant-factor approximation with sublinear worst-case update time is possible. We show that such a lower bound applies even for seemingly simple classes of geometric objects including axis-aligned rectangles in the plane

Assessment criteria for 2D shape transformations in animation

The assessment of 2D shape transformations (or morphing) for animation is a difficult task because it is a multi-dimensional problem. Existing morphing techniques pay most attention to shape information interactive control and mathematical simplicity. This paper shows that it is not enough to use shape information alone, and we should consider other factors such as structure, dynamics, timing, etc. The paper also shows that an overall objective assessment of morphing is impossible because factors such as timing are related to subjective judgement, yet local objective assessment criteria, e.g. based on shape, are available. We propose using “area preservation” as the shape criterion for the 2D case as an acceptable approximation to “volume preservation” in reality, and use it to establish cases in which a number of existing techniques give clearly incorrect results. The possibility of deriving objective assessment criteria for dynamics simulations and timing under certain conditions is discussed

Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph

Data-sensitive metrics adapt distances locally based the density of data points with the goal of aligning distances and some notion of similarity. In this paper, we give the first exact algorithm for computing a data-sensitive metric called the nearest neighbor metric. In fact, we prove the surprising result that a previously published $3$-approximation is an exact algorithm. The nearest neighbor metric can be viewed as a special case of a density-based distance used in machine learning, or it can be seen as an example of a manifold metric. Previous computational research on such metrics despaired of computing exact distances on account of the apparent difficulty of minimizing over all continuous paths between a pair of points. We leverage the exact computation of the nearest neighbor metric to compute sparse spanners and persistent homology. We also explore the behavior of the metric built from point sets drawn from an underlying distribution and consider the more general case of inputs that are finite collections of path-connected compact sets. The main results connect several classical theories such as the conformal change of Riemannian metrics, the theory of positive definite functions of Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop novel proof techniques based on the combination of screw functions and Lipschitz extensions that may be of independent interest.Comment: 15 page