Linear Discrepancy is $\Pi_2$-Hard to Approximate

TL;DR

This paper proves that approximating the linear discrepancy of a matrix within a certain factor is computationally very hard, specifically $ ext{Pi}_2$-hard, strengthening previous NP-hardness results and establishing $ ext{Pi}_2$-completeness.

Contribution

It establishes the $ ext{Pi}_2$-completeness of approximating linear discrepancy, answering an open question and strengthening prior NP-hardness results.

Findings

Approximating linear discrepancy within $9/8 - ext{epsilon}$ is $ ext{Pi}_2$-hard.

The problem is $ ext{Pi}_2$-complete to approximate.

Strengthens previous NP-hardness results for the problem.

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).