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 Π2hard, even to approximate within 9/8 − ǫ factor for any ǫ > 0. This strengthens the NP-hardness result of Li and Nikolov [LN20] 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).

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