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.
Finally, we mention that Klee's measure problem has a very efficient randomized @math -approximation algorithm in time @math with error probability @math @cite_1 .
@cite_1
2137691837 2100228661 1985985425 1569766464
We present a novel analysis of a random sampling approach for four clustering problems in metric spaces: k-median, k-means, min-sum k-clustering, and balanced k-median. For all these problems, we consider the following simple sampling scheme: select a small sample set of input points uniformly at random and then run some approximation algorithm on this sample set to compute an approximation of the best possible clustering of this set. Our main technical contribution is a significantly strengthened analysis of the approximation guarantee by this scheme for the clustering problems.The main motivation behind our analyses was to design sublinear-time algorithms for clustering problems. Our second contribution is the development of new approximation algorithms for the aforementioned clustering problems. Using our random sampling approach, we obtain for these problems the first time approximation algorithms that have running time independent of the input size, and depending on k and the diameter of the metric space only. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006A preliminary extended abstract of this work appeared in Proceedings of the 31st Annual International Colloquium on Automata, Languages and Programming (ICALP), pp. 396-407, 2004. We present a new algorithm for a classic problem in computational geometry, Klee's measure problem: given a set of n axis-parallel boxes in d-dimensional space, compute the volume of the union of the boxes. The algorithm runs in O(nd 2) time for any constant d ≥ 3. Although it improves the previous best algorithm by “just” an iterated logarithmic factor, the real surprise lies in the simplicity of the new algorithm. We also show that it is theoretically possible to beat the O(nd 2) time bound by logarithmic factors for integer input in the word RAM model, and for other variants of the problem. With additional work, we obtain an O(nd 3 polylog n)-time algorithm for the important special case of orthants or unit hypercubes (which include the so-called “hypervolume indicator problem”), and an O(n(d+1) 3 polylog n)-time algorithm for the case of arbitrary hypercubes or fat boxes, improving a previous O(n(d+2) 3)-time algorithm by Bringmann. Given @math elements with nonnegative integer weights @math and an integer capacity @math , we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most the given capacity. We give a deterministic algorithm that estimates the number of solutions to within relative error @math in time polynomial in @math and @math (fully polynomial approximation scheme). More precisely, our algorithm takes time @math . Our algorithm is based on dynamic programming. Previously, randomized polynomial-time approximation schemes were known first by Morris and Sinclair via Markov chain Monte Carlo techniques and subsequently by Dyer via dynamic programming and rejection sampling. Klee’s Measure Problem (KMP) asks for the volume of the union of n axis-aligned boxes in ℝ d . Omitting logarithmic factors, the best algorithm has runtime ( O ^*(n^ d 2 ) ) [Overmars,Yap’91]. There are faster algorithms known for several special cases: Cube-KMP (where all boxes are cubes), Unitcube-KMP (where all boxes are cubes of equal side length), Hypervolume (where all boxes share a vertex), and k -Grounded (where the projection onto the first k dimensions is a Hypervolume instance).