On the complexity and approximability of some Euclidean optimal summing problems

The complexity status of several well-known discrete optimization problems with the direction of optimization switching from maximum to minimum is analyzed. The task is to find a subset of a finite set of Euclidean points (vectors). In these problems, the objective functions depend either only on the norm of the sum of the elements from the subset or on this norm and the cardinality of the subset. It is proved that, if the dimension of the space is a part of the input, then all these problems are strongly $\mathrm{NP}$-hard. Additionally, it is shown that, if the space dimension is fixed, then all the problems are $\mathrm{NP}$-hard even for dimension $2$ (on a plane) and there are no approximation algorithms with a guaranteed accuracy bound for them unless $\mathrm{P=NP}$. It is shown that, if the coordinates of the input points are integer, then all the problems can be solved in pseudopolynomial time in the case of a fixed space dimension.