Complexity and Approximation of the Longest Vector Sum Problem

Abstract

Given a set of n vectors in a d-dimensional normed space, consider the problem of finding a subset with the largest length of the sum vector. We prove that, for any $\ell _p$ norm, $p\in [1,\infty )$, the problem is hard to approximate within a factor better than $\min \{\alpha ^{1/p},\sqrt{\alpha }\}$, where $\alpha =16{\text{/ }}17$. In the general case, we show that the cardinality-constrained version of the problem is hard for approximation factors better than $1-1/e$ and is W[2]-hard with respect to the cardinality of the solution. For both original and cardinality-constrained problems, we propose a randomized $(1-\varepsilon )$-approximation algorithm that runs in polynomial time when the dimension of space is $O(\log n)$. The algorithm has a linear time complexity for any fixed d and $\varepsilon \in (0,1)$.