On a Quadratic Euclidean Problem of Vector Subset Choice: Complexity and Algorithmic Approach

We analyze the complexity status of one of the known discrete optimization problems where the optimization criterium is switched from max to min. In the considered problem, we search in a finite set of Euclidean vectors (points) a subset that minimizes the squared norm of the sum of its elements divided by the cardinality of the subset. It is proved that if the dimension of the space is a part of input then the problem is NP-hard in a strong sense. Also, if the dimension of the space is fixed then the problem is NP-hard even for dimension 1 (on a line) and there are no approximation algorithms with guaranteed approximation ratio unless P=NP. It is shown that if the coordinates of the input vectors are integer then even a more general problem can be solved in a pseudopolynomial time in case when the dimension of the space is fixed.