On two problems of choosing some subset of vectors with integer coordinates that has maximum norm of the sum of elements in Euclidean space

Abstract

Under study are the two problems of choosing a subset of m vectors with the maximum norm of the sum of the elements from a set of n vectors in Euclidean space ℝ k . The vectors are assumed to have integer coordinates. Using the dynamic programming technique, some new optimal algorithms are suggested for solving the problems; these algorithms have pseudopolynomial time when the dimension of the space is fixed. The new algorithms have certain advantages over the availables: the vector subset problem can be solved faster for m < (k/2) k , while, after taking into account an additional restriction on the order of the vectors, the time complexity is k k−1 times less independently on m.