Number Balancing is as Hard as Minkowski's Theorem and Shortest Vector

The number balancing (NBP) problem is the following: given real numbers \(a_1,\ldots ,a_n \in [0,1]\), find two disjoint subsets \(I_1,I_2 \subseteq [n]\) so that the difference \(|\sum _{i \in I_1}a_i - \sum _{i \in I_2}a_i|\) of their sums is minimized. An application of the pigeonhole principle shows that there is always a solution where the difference is at most \(O(\frac{\sqrt{n}}{2^n})\). Finding the minimum, however, is NP-hard. In polynomial time, the differencing algorithm by Karmarkar and Karp from 1982 can produce a solution with difference at most \(n^{-\varTheta (\log n)}\), but no further improvement has been made since then.