In this note, we prove that the problem of computing the linear discrepancy of a given matrix is $\Pi_2$-hard, even to approximate within $9/8 - \epsilon$ factor for any $\epsilon > 0$. This strengthens the NP-hardness result of Li and Nikolov [ESA 2020] for the exact version of the problem, and answers a question posed by them. Furthermore, since Li and Nikolov showed that the problem is contained in $\Pi_2$, our result makes linear discrepancy another natural problem that is $\Pi_2$-complete (to approximate).

翻译：在本说明中,我们证明计算某一矩阵线性差异的问题在于$\Pi_2$-hard,甚至近似于9/8 -\ epsilon$(美元) > 0美元。这强化了Li和Nikolov(ESA 2020)对问题确切版本的NP-hard性结果,并回答了他们提出的问题。此外,由于Li和Nikolov显示问题包含在$\Pi_2美元中,我们的结果使得线性差异成为另一个自然问题,即$\Pi_2$(约合)。