Linear Discrepancy is $\Pi_2$-Hard to Approximate

Pasin Manurangsi

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arXiv · 2107.01235

Linear Discrepancy is $\Pi_2$-Hard to Approximate

Pasin Manurangsi

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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).

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BibTeX RIS

Pasin Manurangsi. 2021-07-02. Linear Discrepancy is $\Pi_2$-Hard to Approximate. https://arxiv.org/abs/2107.01235

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