An approximation algorithm for a problem of partitioning a sequence into clusters with restrictions on their cardinalities

Abstract

We consider the problem of partitioning a finite sequence of points in Euclidean space into a given number of clusters (subsequences) minimizing the sum of squared distances between cluster elements and the corresponding cluster centers. It is assumed that the center of one of the desired clusters is the origin, while the centers of the other clusters are unknown and determined as the mean values over clusters elements. Additionally, there are a few structural restrictions on the elements of clusters with unknown centers: (1) clusters form non-overlapping subsequences of the input sequence, (2) the difference between two consecutive indices is bounded from below and above by prescribed constants, and (3) the total number of elements in these clusters is given as an input. It is shown that the problem is strongly NP-hard. A 2-approximation algorithm which runs in polynomial time for a fixed number of clusters is proposed for this problem.