The problem of finding a subset of vectors with the maximum total weight

The NP-hardness is proved for the discrete optimization problems connected with maximizing the total weight of a subset of a finite set of vectors in Euclidean space, i.e., the Euclidean norm of the sum of the members. Two approximation algorithms are suggested, and the bounds for the relative error and time complexity are obtained. We give a polynomial approximation scheme in the case of a fixed dimension and distinguished a subclass of problems solvable in pseudopolynomial time. The results obtained are useful for solving the problem of choice of a fixed number of subsequences from a numerical sequence with a given number of quasiperiodical repetitions of the subsequence.