A Fully Polynomial-Time Approximation Scheme for a Special Case of a Balanced 2-Clustering Problem
We consider the strongly NP-hard problem of partitioning a set of Euclidean points into two clusters so as to minimize the sum (over both clusters) of the weighted sum of the squared intracluster distances from the elements of the clusters to their centers. The weights of sums are the cardinalities of the clusters. The center of one of the clusters is given as input, while the center of the other cluster is unknown and determined as the geometric center (centroid), i.e. the average value over all points in the cluster. We analyze the variant of the problem with cardinality constraints. We present an approximation algorithm for the problem and prove that it is a fully polynomial-time approximation scheme when the space dimension is bounded by a constant.
KeywordsNP-hardness Euclidian space Fixed dimension FPTAS
This work was supported by the RFBR, projects 15-01-00462, 16-31-00186 and 16-07-00168.
- 7.Kel’manov, A.V., Motkova, A.V.: An exact pseudopolynomial algorithm for a special case of a euclidean balanced variance-based 2-clustering problem. In: Abstracts of the VI International Conference “Optimization and Applications” (OPTIMA-2015), P. 98. Petrovac, Montenegro (2015)Google Scholar
- 10.Inaba, M., Katoh, N., Imai, H.: Applications of Weighted Voronoi Diagrams and Randomization toVariance-Based \(k\)-Clustering: (extended abstract). Stony Brook, NY, USA, pp. 332–339 (1994)Google Scholar
- 11.Hasegawa, S., Imai, H., Inaba, M., Katoh, N., Nakano, J.: Efficient algorithms for variance-based \(k\)-clustering. In: Proceedings of the 1st Pacific Conference on Computer Graphics andApplications (Pacific Graphics 1993, Seoul, Korea),World Scientific, River Edge, NJ. 1, pp. 75–89 (1993)Google Scholar
- 13.de la Vega, F., Karpinski, M., Kenyon, C., Rabani, Y.: Polynomial Time Approximation Schemes for Metric Min-Sum Clustering. Electronic Colloquium on Computational Complexity (ECCC), 25 (2002)Google Scholar