max planck institut
informatik
informatik
Max-Planck-Institut für Informatik
Department 1: Algorithms and Complexity
Campus E1 4, Room 322
66123 Saarbrücken
Germany
Email:
karl.bringmann
mpi-inf.mpg.de
Phone: +49 681 9325 1022
Fax: +49 681 9325 1099
For a complete list of publications look here
We study the fundamental problem of the exact and efficient generation of random values from a finite and discrete probability distribution. Suppose that we are given n distinct events with associated probabilities p_1,...,p_n. We consider the problem of sampling a subset, which includes the i-th event independently with probability p_i, and the problem of sampling from the distribution, where the i-th event has a probability proportional to p_i. For both problems we present on two different classes of inputs – sorted and general probabilities – efficient preprocessing algorithms that allow for asymptotically optimal querying, and prove almost matching lower bounds for their complexity.
Let P be a set of n points in the plane. A crossing-free structure on P is a straight-edge planar graph with vertex set in P. Examples of crossing-free structures include triangulations of P, and spanning cycles of P, also known as polygonalizations of P, among others. There has been a large amount of research trying to bound the number of such structures. In particular, bounding the number of triangulations spanned by P has received considerable attention. It is currently known that every set of n points has at most O(30n) and at least Ω(2.43n) triangulations. However, much less is known about the algorithmic problem of counting crossing-free structures of a given set P. For example, no algorithm for counting triangulations is known that, on all instances, performs faster than enumerating all triangulations. In this paper we develop a general technique for computing the number of crossing-free structures of an input set P. We apply the technique to obtain algorithms for computing the number of triangulations and spanning cycles of P. The running time of our algorithms is upper bounded by nO(k), where k is the number of onion layers of P. In particular, we show that our algorithm for counting triangulations is not slower than O(3.1414n). Given that there are several well-studied configurations of points with at least Ω(3.464n) triangulations, and some even with Ω(8n) triangulations, our algorithm is the first to asymptotically outperform any enumeration algorithm for such instances. In fact, it is widely believed that any set of n points must have at least Ω(3.464n) triangulations. If this is true, then our algorithm is strictly sub-linear in the number of triangulations counted. We also show that our techniques are general enough to solve the restricted triangulation counting problem, which we prove to be W[2]-hard in the parameter k. This implies a “no free lunch” result: In order to be fixed-parameter tractable, our general algorithm must rely on additional properties that are specific to the considered class of structures.
In order to allow a comparison of (otherwise incomparable) sets, many evolutionary multiobjective optimizers use indicator functions to guide the search and to evaluate the performance of search algorithms. The most widely used indicator is the hypervolume indicator. It measures the volume of the dominated portion of the objective space.
Though the hypervolume indicator is very popular, it has not been shown that maximizing the hypervolume indicator is indeed equivalent to the overall objective of finding a good approximation of the Pareto front. To address this question, we compare the optimal approximation factor with the approximation factor achieved by sets maximizing the hypervolume indicator. We bound the optimal approximation factor of n points by 1+Θ(1/n) for arbitrary Pareto fronts. Furthermore, we prove that the same asymptotic approximation ratio is achieved by sets of n points that maximize the hypervolume indicator. This shows that the speed of convergence of the approximation ratio achieved by maximizing the hypervolume indicator is asymptotically optimal.
This implies that for large values of n, sets maximizing the hypervolume indicator quickly approach the optimal approximation ratio. Moreover, our bounds show that also for relatively small values of n, sets maximizing the hypervolume indicator achieve a near-optimal approximation ratio.
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 Klee's 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 2d1-ε-approximation for certain boxes unless NP=BPP, but give a simple additive polynomial-time ε-approximation.
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