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

Primal-Dual Approximation Algorithms for Metric Facility Location and k-Median Problems

We present approximation algorithms for the metric uncapacitated facility location problem and the metric k-median problem achieving guarantees of 3 and 6 respectively. The distinguishing feature of our algorithms is their low running time: O(m log m) and O(m logm(L+log(n))) respectively, where n and m are the total number of vertices and edges in the underlying graph. The main algorithmic idea...

Segmentation of Photospheric Solar Images by Using c-Means,k-Means, and FCM Algorithms

Interpolating between k-Median and k-Center: Approximation Algorithms for Ordered k-Median

We consider a generalization of k-median and k-center, called the ordered k-median problem. In this problem, we are given a metric space (D, {cij}) with n = |D| points, and a non-increasing weight vector w ∈ Rn+, and the goal is to open k centers and assign each point each point j ∈ D to a center so as to minimize w1 · (largest assignment cost) + w2 · (second-largest assignment cost) + . . . + ...

Distributed k-Means and k-Median Clustering on General Topologies

This paper provides new algorithms for distributed clustering for two popular center-based objectives, k-median and k-means. These algorithms have provable guarantees and improve communication complexity over existing approaches. Following a classic approach in clustering by [13], we reduce the problem of finding a clustering with low cost to the problem of finding a coreset of small size. We p...

Approximation Algorithms for Aversion k-Clustering via Local k-Median

In the aversion k-clustering problem, given a metric space, we want to cluster the points into k clusters. The cost incurred by each point is the distance to the furthest point in its cluster, and the cost of the clustering is the sum of all these per-point-costs. This problem is motivated by questions in generating automatic abstractions of extensive-form games. We reduce this problem to a “lo...