An Approximation Scheme for a Weighted Two-Cluster Partition Problem
We consider the problem of partitioning a set of Euclidean points into two clusters to minimize the weighted sum of the squared intracluster distances from the elements of the clusters to their centers. The center of one of the clusters is unknown and determined as the average value over all points in the cluster, while the center of the other cluster is the origin. The weight factors for both intracluster sums are given as input. We present an approximation algorithm for the problem, which is based on an adaptive-grid-approach. The algorithm implements a fully polynomial-time approximation scheme (FPTAS) in the case of the fixed space dimension. In the case when the dimension of space is not fixed but is bounded by a slowly growing function of the number of input points, the algorithm realizes a polynomial-time approximation scheme (PTAS).
KeywordsWeighted 2-clustering NP-hardness Euclidean space FPTAS PTAS
This work was supported by the Russian Science Foundation (project 16-11-10041).
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