Approximation Algorithms for Clustering with Dynamic Points
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In many classic clustering problems, we seek to sketch a massive data set of n points in a metric space, by segmenting them into k categories or clusters, each cluster represented concisely by a single point in the metric space. Two notable examples are the k-center/k-supplier problem and the k-median problem. In practical applications of clustering, the data set may evolve over time, reflecting an evolution of the underlying clustering model. In this paper, we initiate the study of a dynamic version of clustering problems that aims to capture these considerations. In this version there are T time steps, and in each time step t∈{1,2,…,T}, the set of clients needed to be clustered may change, and we can move the k facilities between time steps. More specifically, we study two concrete problems in this framework: the Dynamic Ordered k-Median and the Dynamic k-Supplier problem. We first consider the Dynamic Ordered k-Median problem, where the objective is to minimize the weighted sum of ordered distances over all time steps, plus the total cost of moving the facilities between time steps. We present one constant-factor approximation algorithm for T=2 and another approximation algorithm for fixed T ≥ 3. Then we consider the Dynamic k-Supplier problem, where the objective is to minimize the maximum distance from any client to its facility, subject to the constraint that between time steps the maximum distance moved by any facility is no more than a given threshold. When the number of time steps T is 2, we present a simple constant factor approximation algorithm and a bi-criteria constant factor approximation algorithm for the outlier version, where some of the clients can be discarded. We also show that it is NP-hard to approximate the problem with any factor for T ≥ 3.
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