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

TL;DR

This paper establishes that the Minimum Distance Problem over all finite fields and the Shortest Vector Problem in all  norms are W[1]-hard to approximate within any constant factor, resolving key open questions in parameterized complexity.

Contribution

It proves W[1]-hardness of approximation for MDP over all fields and SVP in all  norms, extending previous results and answering open questions.

Findings

MDP is W[1]-hard to approximate within any constant factor over all finite fields.

SVP in  norms is W[1]-hard to approximate within any constant factor for p>1.

SVP in  norms is W[1]-hard to approximate within a factor approaching 2 for p=1.

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.