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

Abstract

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∈[1,2] and is not in APX if p∈(2,∞). In the case of an arbitrary norm, we propose an algorithm which finds an optimal solution in time O(nd−1(d+log⁡n)), 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 O(dnd+1). In the two-dimensional case, this version is shown to be solvable in nearly quadratic time.