Linear Discrepancy is Π2-Hard to Approximate

In this note, we prove that the problem of computing the linear discrepancy of a given matrix is Π2- hard, even to approximate within 9/8 − ε factor for any ε > 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 Π2, our result makes linear discrepancy another natural problem that is Π2-complete (to approximate).

• Algorithms and Theory