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We consider the computation of the volume of the union of high-dimensional geometric objects. While showing that this problem is #P-hard already for very simple bodies (i.e., axis-parallel boxes), we give a fast FPRAS for all objects where one can: (1) test whether a given point lies inside the object, (2) sample a point uniformly, (3) calculate the volume of the object in polynomial time. All three oracles can be weak, that is, just approximate. This implies that Klees measure problem and the hypervolume indicator can be approximated efficiently even though they are #P-hard and hence cannot be solved exactly in time polynomial in the number of dimensions unless P=NP. Our algorithm also allows to approximate efficiently the volume of the union of convex bodies given by weak membership oracles. For the analogous problem of the intersection of high-dimensional geometric objects we prove #P-hardness for boxes and show that there is no multiplicative polynomial-time $2^{d^{1-epsilon}}$-approximation for certain boxes unless NP=BPP, but give a simple additive polynomial-time $epsilon$-approximation.
Compressed bitmap indexes are used to speed up simple aggregate queries in databases. Indeed, set operations like intersections, unions and complements can be represented as logical operations (AND,OR,NOT) that are ideally suited for bitmaps. However
We give a polynomial-time constant-factor approximation algorithm for maximum independent set for (axis-aligned) rectangles in the plane. Using a polynomial-time algorithm, the best approximation factor previously known is $O(loglog n)$. The results
Let $P$ be a set of points in $mathbb{R}^d$, $B$ a bicoloring of $P$ and $Oo$ a family of geometric objects (that is, intervals, boxes, balls, etc). An object from $Oo$ is called balanced with respect to $B$ if it contains the same number of points f
The Euclidean $k$-center problem is a classical problem that has been extensively studied in computer science. Given a set $mathcal{G}$ of $n$ points in Euclidean space, the problem is to determine a set $mathcal{C}$ of $k$ centers (not necessarily p
Graph drawing addresses the problem of finding a layout of a graph that satisfies given aesthetic and understandability objectives. The most important objective in graph drawing is minimization of the number of crossings in the drawing, as the aesthe