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Sublineartime approximation for clustering via random sampling
 In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP’04
, 2004
"... Abstract. In this paper we present a novel analysis of a random sampling approach for three clustering problems in metric spaces: kmedian, minsum kclustering, and balanced kmedian. For all these problems we consider the following simple sampling scheme: select a small sample set of points unifor ..."
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Abstract. In this paper we present a novel analysis of a random sampling approach for three clustering problems in metric spaces: kmedian, minsum kclustering, and balanced kmedian. For all these problems we consider the following simple sampling scheme: select a small sample set of points uniformly at random from V 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 sublineartime 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 the first time approximation algorithms that have the running time independent of the input size, and depending on k and the diameter of the metric space only. 1
A Polynomial Time Approximation Scheme for Subdense MAXCUT
 Electronic Colloquium on Computational Complexity
, 2002
"... We prove that the subdense instances of MAXCUT of average degree Ω(n/log n) possesses a polynomial time approximation scheme (PTAS). We extend this result also to show that the instances of general 2ary maximum constraint satisfaction problems (MAX2CSP) of the same average density have ..."
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Cited by 3 (0 self)
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We prove that the subdense instances of MAXCUT of average degree &Omega;(n/log n) possesses a polynomial time approximation scheme (PTAS). We extend this result also to show that the instances of general 2ary maximum constraint satisfaction problems (MAX2CSP) of the same average density have PTASs. Our results display for the first time an existence of PTASs for these subdense classes.
Approximation Schemes for Clustering Problems in Finite Metrics and High Dimensional Spaces
, 2002
"... We give polynomial time approximation schemes (PTASs) for the problem of partitioning an input set of n points into a fixed number k of clusters so as to minimize the sum over all clusters of the sum of pairwise distances in a cluster. Our algorithms work for arbitrary metric spaces as well as for p ..."
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We give polynomial time approximation schemes (PTASs) for the problem of partitioning an input set of n points into a fixed number k of clusters so as to minimize the sum over all clusters of the sum of pairwise distances in a cluster. Our algorithms work for arbitrary metric spaces as well as for points in R^d where the distance between two points x, y is measured by kx yk 2 (notice that (R^d, k k 2 ) is not a metric space). Our algorithms can be modified to handle other objective functions, such as minimizing the sum over all clusters of the sum of distances to the best choice for a cluster center. The method of solution of this paper depends on some new techniques which could be also of independent interest.
Approximation Schemes for Clustering Problems (Extended Abstract)
 STOC'03
, 2003
"... Let k be a fixed integer. We consider the problem of partitioning an input set of points endowed with a distance function into k clusters. We give polynomial time approximation schemes for the following three clustering problems: Metric kClustering, ` 2 kClustering, and ` 2 kMedian. In the kCl ..."
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Let k be a fixed integer. We consider the problem of partitioning an input set of points endowed with a distance function into k clusters. We give polynomial time approximation schemes for the following three clustering problems: Metric kClustering, ` 2 kClustering, and ` 2 kMedian. In the kClustering problem, the objective is to minimize the sum of all intracluster distances. In the kMedian problem, the goal is to minimize the sum of distances from points in a cluster to the (best choice of) cluster center. In metric instances, the input distance function is a metric. In ` 2 instances, the points are in R and the distance between two points x; y is measured by kx \Gamma yk 2 (notice that (R ; k \Delta k 2 ) is not a metric space). For the first two problems, our results are the first polynomial time approximation schemes. For the third problem, the running time of our algorithms is a vast improvement over previous work.