Approximation Algorithms for Min-Sum k-Clustering and Balanced k-Median

We consider two closely related fundamental clustering problems in this paper. In the min-sum k-clustering one is given a metric space and has to partition the points into k clusterswhile minimizing the sum of pairwise distances between the points within the clusters. In theBalanced k-Median problem the instance is the same and one has to obtain a clustering into kcluster C1, . . . , Ck, where each cluster Ci has a center ci, while minimizing the total assignmentcosts for the points in the metric; here the cost of assigning a point j to a cluster Ci is equal to|Ci| times the j, cj distance in the metric.In this paper, we present an O(log n)-approximation for both these problems where n isthe number of points in the metric that are to be served. This is an improvement over theO( −1 log n)-approximation (for any constant > 0) obtained by Bartal, Charikar, and Raz[STOC ’01]. We also obtain a quasi-PTAS for Balanced k-Median in metrics with constantdoubling dimension.As in the work of Bartal et al., our approximation for general metrics uses embeddingsinto tree metrics. The main technical contribution in this paper is an O(1)-approximation forBalanced k-Median in hierarchically separated trees (HSTs). Our improvement comes from amore direct dynamic programming approach that heavily exploits properties of standard HSTs.In this way, we avoid the reduction to special types of HSTs that were considered by Bartal etal., thereby avoiding an additional O( −1 log n) loss. ∗Email: [email protected].†Email: [email protected].‡Supported by NSERC. Email: [email protected].§Email: [email protected].