A bi-criteria approximation algorithm for kk Means

We consider the classical $k$-means clustering problem in the setting bi-criteria approximation, in which an algoithm is allowed to output $\beta k > k$ clusters, and must produce a clustering with cost at most $\alpha$ times the to the cost of the optimal set of $k$ clusters. We argue that this approach is natural in many settings, for which the exact number of clusters is a priori unknown, or unimportant up to a constant factor. We give new bi-criteria approximation algorithms, based on linear programming and local search, respectively, which attain a guarantee $\alpha(\beta)$ depending on the number $\beta k$ of clusters that may be opened. Our gurantee $\alpha(\beta)$ is always at most $9 + \epsilon$ and improves rapidly with $\beta$ (for example: $\alpha(2)<2.59$, and $\alpha(3) < 1.4$). Moreover, our algorithms have only polynomial dependence on the dimension of the input data, and so are applicable in high-dimensional settings

Maximum gradient embeddings and monotone clustering

Let (X,d_X) be an n-point metric space. We show that there exists a distribution D over non-contractive embeddings into trees f:X-->T such that for every x in X, the expectation with respect to D of the maximum over y in X of the ratio d_T(f(x),f(y)) / d_X(x,y) is at most C (log n)^2, where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location.Comment: 25 pages, 2 figures. Final version, minor revision of the previous one. To appear in "Combinatorica

Fast Clustering with Lower Bounds: No Customer too Far, No Shop too Small

We study the \LowerBoundedCenter (\lbc) problem, which is a clustering problem that can be viewed as a variant of the \kCenter problem. In the \lbc problem, we are given a set of points P in a metric space and a lower bound \lambda, and the goal is to select a set C \subseteq P of centers and an assignment that maps each point in P to a center of C such that each center of C is assigned at least \lambda points. The price of an assignment is the maximum distance between a point and the center it is assigned to, and the goal is to find a set of centers and an assignment of minimum price. We give a constant factor approximation algorithm for the \lbc problem that runs in O(n \log n) time when the input points lie in the d-dimensional Euclidean space R^d, where d is a constant. We also prove that this problem cannot be approximated within a factor of 1.8-\epsilon unless P = \NP even if the input points are points in the Euclidean plane R^2.Comment: 14 page

Better Guarantees for k-Means and Euclidean k-Median by Primal-Dual Algorithms

bibsource: dblp computer science bibliography, http://dblp.org biburl: http://dblp.org/rec/bib/conf/focs/AhmadianNSW17 timestamp: Thu, 16 Nov 2017 15:01:42 +0100 bdsk-url-1: https://doi.org/10.1109/FOCS.2017.15 bdsk-url-2: http://dx.doi.org/10.1109/FOCS.2017.15bibsource: dblp computer science bibliography, http://dblp.org biburl: http://dblp.org/rec/bib/conf/focs/AhmadianNSW17 timestamp: Thu, 16 Nov 2017 15:01:42 +0100 bdsk-url-1: https://doi.org/10.1109/FOCS.2017.15 bdsk-url-2: http://dx.doi.org/10.1109/FOCS.2017.1