Complexity and algorithms for finding a subset of vectors with the longest sum

The problem is, given a set of n vectors in a d-dimensional normed space, find a subset with the largest length of the sum vector. We prove that, in the case of the lp norm, the problem is APX-complete for any p is an element of [1, 2] and is not in APX if p is an element of (2, infinity). In the case of an arbitrary norm, we propose an algorithm which finds an optimal solution in time 0 (n(d-1)(d + logn)), improving previously known algorithms. In particular, the two-dimensional problem can be solved in nearly linear time. We also present an improved algorithm for the cardinality-constrained version of the problem with running time 0 (dn(d+1)). In the two-dimensional case, this version is shown to be solvable in nearly quadratic time. (C) 2018 Elsevier B.V. All rights reserved.