Parameterized inapproximability of the minimum distance problem over all fields and the shortest vector problem in all $\ell_p$ norms. Bennett, H., Cheraghchi, M., Guruswami, V., & Ribeiro, J. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing (STOC), 2023. ![Parameterized inapproximability of the minimum distance problem over all fields and the shortest vector problem in all $\ell_p$ norms [link]](https://bibbase.org/img/filetypes/link.svg "link")
Paper abstract bibtex We prove that the Minimum Distance Problem (MDP) on linear codes over any fixed finite field and parameterized by the input distance bound is W[1]-hard to approximate within any constant factor. We also prove analogous results for the parameterized Shortest Vector Problem (SVP) on integer lattices. Specifically, we prove that SVP in the $\ell_p$ norm is W[1]-hard to approximate within any constant factor for any fixed $p > 1$ and W[1]-hard to approximate within a factor approaching 2 for $p = 1$. (These results all hold under randomized reductions.) These results answer the main questions left open (and asked) by Bhattacharyya, Bonnet, Egri, Ghoshal, Karthik C. S., Lin, Manurangsi, and Marx (Journal of the ACM, 2021) on the complexity of parameterized MDP and SVP. For MDP, they established similar hardness for binary linear codes and left the case of general fields open. For SVP in $\ell_p$ norms with $p > 1$, they showed inapproximability within some constant factor (depending on $p$) and left open showing such hardness for arbitrary constant factors. They also left open showing W[1]-hardness even of exact SVP in the $\ell_1$ norm.
@INPROCEEDINGS{ref:BCGR23,
author = {Huck Bennett and Mahdi Cheraghchi and Venkatesan Guruswami and Jo\~{a}o Ribeiro},
title = {Parameterized inapproximability of the minimum distance problem over all fields and the shortest vector problem in all $\ell_p$ norms},
booktitle = {Proceedings of the 55th Annual ACM Symposium on Theory of Computing (STOC)},
year = 2023,
url_Paper = {https://eccc.weizmann.ac.il/report/2022/156/},
abstract = {We prove that the Minimum Distance Problem (MDP) on linear codes over any fixed finite field and parameterized by the input distance bound is W[1]-hard to approximate within any constant factor. We also prove analogous results for the parameterized Shortest Vector Problem (SVP) on integer lattices. Specifically, we prove that SVP in the $\ell_p$ norm is W[1]-hard to approximate within any constant factor for any fixed $p > 1$ and W[1]-hard to approximate within a factor approaching 2 for $p = 1$. (These results all hold under randomized reductions.)
These results answer the main questions left open (and asked) by Bhattacharyya, Bonnet, Egri, Ghoshal, Karthik C. S., Lin, Manurangsi, and Marx (Journal of the ACM, 2021) on the complexity of parameterized MDP and SVP.
For MDP, they established similar hardness for binary linear codes and left the case of general fields open. For SVP in $\ell_p$ norms with $p > 1$, they showed inapproximability within some constant factor (depending on $p$) and left open showing such hardness for arbitrary constant factors. They also left open showing W[1]-hardness even of exact SVP in the $\ell_1$ norm.}
}
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