Linear Discrepancy is ${\Pi }_2$-Hard to Approximate

Abstract

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).

arXiv comment9 pages; to appear in Information Processing Letters