On the Gap between Hereditary Discrepancy and the Determinant Lower Bound
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929211469463552 |
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| author | Li, Lily Nikolov, Aleksandar |
| author_facet | Li, Lily Nikolov, Aleksandar |
| contents | The determinant lower bound of Lovasz, Spencer, and Vesztergombi [European Journal of Combinatorics, 1986] is a powerful general way to prove lower bounds on the hereditary discrepancy of a set system. In their paper, Lovasz, Spencer, and Vesztergombi asked if hereditary discrepancy can also be bounded from above by a function of the hereditary discrepancy. This was answered in the negative by Hoffman, and the largest known multiplicative gap between the two quantities for a set system of $m$ substes of a universe of size $n$ is on the order of $\max\{\log n, \sqrt{\log m}\}$. On the other hand, building on work of Matoušek [Proceedings of the AMS, 2013], recently Jiang and Reis [SOSA, 2022] showed that this gap is always bounded up to constants by $\sqrt{\log(m)\log(n)}$. This is tight when $m$ is polynomial in $n$, but leaves open what happens for large $m$. We show that the bound of Jiang and Reis is tight for nearly the entire range of $m$. Our proof relies on a technique of amplifying discrepancy via taking Kronecker products, and on discrepancy lower bounds for a set system derived from the discrete Haar basis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_08167 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Gap between Hereditary Discrepancy and the Determinant Lower Bound Li, Lily Nikolov, Aleksandar Combinatorics Discrete Mathematics The determinant lower bound of Lovasz, Spencer, and Vesztergombi [European Journal of Combinatorics, 1986] is a powerful general way to prove lower bounds on the hereditary discrepancy of a set system. In their paper, Lovasz, Spencer, and Vesztergombi asked if hereditary discrepancy can also be bounded from above by a function of the hereditary discrepancy. This was answered in the negative by Hoffman, and the largest known multiplicative gap between the two quantities for a set system of $m$ substes of a universe of size $n$ is on the order of $\max\{\log n, \sqrt{\log m}\}$. On the other hand, building on work of Matoušek [Proceedings of the AMS, 2013], recently Jiang and Reis [SOSA, 2022] showed that this gap is always bounded up to constants by $\sqrt{\log(m)\log(n)}$. This is tight when $m$ is polynomial in $n$, but leaves open what happens for large $m$. We show that the bound of Jiang and Reis is tight for nearly the entire range of $m$. Our proof relies on a technique of amplifying discrepancy via taking Kronecker products, and on discrepancy lower bounds for a set system derived from the discrete Haar basis. |
| title | On the Gap between Hereditary Discrepancy and the Determinant Lower Bound |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2303.08167 |