Linear discrepancy is Π2-hard to approximate

Abstract

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 [9] 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).