NP-hardness of some Quadratic Euclidean 2-clustering problems

作者：Kel'manov A V^*; Pyatkin A V

来源：Doklady Mathematics, 2015, 92(2): 634-637.

DOI：10.1134/S1064562415050233

Some problems of partitioning a finite set of points of Euclidean space into two clusters are considered. In these problems, the following criteria are minimized: (1) the sum over both clusters of the sums of squared pairwise distances between the elements of the cluster and (2) the sum of the (multiplied by the cardinalities of the clusters) sums of squared distances from the elements of the cluster to its geometric center. Additionally, another problem close to (2) is considered, in which the desired center of one of the clusters is given as input, while the center of the other cluster is unknown (is the variable to be optimized) as in problem (2). Two variants of the problems are analyzed, in which the cardinalities of the clusters are parts of the input or optimization variables. It is proved that all the considered problems are strongly NP-hard and that, in general, there is no fully polynomial-time approximation scheme for them (unless P = NP).

• 出版日期2015-9