Given two sets of numbers: $$A = \{a_1,a_2,...,a_n\}, B=\{b_1,b_2,...,b_n\}$$ and assume the sumation $\sum_i a_i = \sum_i b_i =0$ vanish. How to estimate the minimum $s$ in the following?
$$s = \min_{\sigma} |S(\sigma)| \equiv \min_{\sigma} \Big| \sum_{i=1}^n a_i b_{\sigma(i)} \Big|,$$ where $\sigma$ is a permutation of $\{1,2,...,n\}$.
Furthermore, how to characterize the probability distribution of $S(\sigma)$?
PS: The question becomes simple if we remove the absolute value in $S$, as we have the rearrangement inequality.