Complexity and Approximation of Finding the Longest Vector Sum

Abstract

The problem under study is, given a finite set of vectors in a normed vector space, find a subset which maximizes the norm of the vector sum. For each $${{\ell }_{p}}$$ norm, $$p \in [1,\infty )$$ , the problem is proved to have an inapproximability bound in the class of polynomial-time algorithms. For an arbitrary normed space of dimension $$O(logn)$$ , a randomized polynomial-time approximation scheme is proposed.