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Revision #2 to TR22-156 | 27th February 2024 03:29

Parameterized Inapproximability of the Minimum Distance Problem over all Fields and the Shortest Vector Problem in all $\ell_p$ Norms

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Revision #2
Authors: Huck Bennett, Mahdi Cheraghchi, Venkatesan Guruswami, Joao Ribeiro
Accepted on: 27th February 2024 03:29
Downloads: 89
Keywords: 


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 $p=1$.(We show hardness under randomized reductions in each case.)

These results answer the main questions left open (and explicitly posed) 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.



Changes to previous version:

Improved exposition. To appear at SICOMP.


Revision #1 to TR22-156 | 25th February 2023 01:28

Parameterized Inapproximability of the Minimum Distance Problem over all Fields and the Shortest Vector Problem in all $\ell_p$ Norms





Revision #1
Authors: Huck Bennett, Mahdi Cheraghchi, Venkatesan Guruswami, Joao Ribeiro
Accepted on: 25th February 2023 01:28
Downloads: 222
Keywords: 


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 $p=1$.(We show hardness under randomized reductions in each case.)

These results answer the main questions left open (and explicitly posed) 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.



Changes to previous version:

Implemented reviewers' comments. To appear at STOC 2023.


Paper:

TR22-156 | 15th November 2022 20:52

Parameterized Inapproximability of the Minimum Distance Problem over all Fields and the Shortest Vector Problem in all $\ell_p$ Norms


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 $p=1$.(We show hardness under randomized reductions in each case.)

These results answer the main questions left open (and explicitly posed) 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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