Fast Approximation Algorithms for Some Maximin Clustering Problems

Abstract

In this paper, we consider three cases of an intractable problem of searching for two subsets in a finite set of points of Euclidean space. In all three cases, it is required to maximize the minimum cluster’s cardinality under constraint on each cluster’s scatter. The scatter is the sum of the distances from the cluster elements to the center, which is defined differently in each of the three cases. In the first case, cluster centers are fixed points. In the second case, the centers are unknown points from the input set. In the third case, the centers are defined as the centroids (the arithmetic mean) of clusters. We propose a general scheme that allows us to build a polynomial 1/2-approximation algorithm for a generalized problem and can be used for constructing 1/2-approximation algorithms for the first two cases and for the one-dimensional third case. Also we show how, using precomputed general information, their time complexities can be reduced to the complexity of sorting. Finally, we present the results of computational experiments showing the accuracy of the proposed algorithms on randomly generated input data.