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 ${\ell }_p$ norm, the problem is APX-complete for any $p\in [1,2]$ and is not in APX if $p\in (2,\infty )$. In the case of an arbitrary norm, we propose an algorithm which finds an optimal solution in time $O(n^{d-1}(d+\log n))$, 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 $O(d{n}^{d+1})$. In the two-dimensional case, this version is shown to be solvable in nearly quadratic time.

问题是，给定d维范数空间中的一组n个向量，找到总和向量长度最大的子集。我们证明，对于${\ell }_p$ 规范，任何问题都是APX完整的 $p\in [\mathrm{1个}，2]$ 并且不在APX中，如果 $p\in （2，\infty ）$。在任意范数的情况下，我们提出一种算法，该算法可以及时找到最优解$Ø（{ñ}^{d-\mathrm{1个}}（d+\mathrm{日志}ñ））$，改进了以前已知的算法。特别地，二维问题可以在几乎线性的时间内解决。我们还针对带有运行时间的问题的基数受限版本提出了一种改进的算法$Ø（d{ñ}^{d+\mathrm{1个}}）$。在二维情况下，该版本可在几乎二次时间内解决。