Hardness of checking maximal magnitude of a sum of a subset of vectors
I suppose you are talking about the euclidean norm. Consider the optimal solution $S$ and let $w=\sum_{v\in S}v$, and we assume there's no zero vector in $V$.

For any $v\notin S$, it must holds $v\cdot w&lt;0$, otherwise $\|w+v\|^2\geq \|w\|^2+\|v\|^2&gt;\|w\|^2$. Similarly for $v\in S$ it holds $v\cdot w&gt;0$, otherwise $\|w-v\|&gt;\|w\|$.

That means $S$ is exactly determined by the direction of $w$. You can exhaust the possibilities by sorting the polar angles of the vectors, and rotate the orthogonal line of $w$ which has only $n$ essentially different partitions.