论文标题
所有字段上最小距离问题的参数化不Xibibibibibibibibibibibibibibibibibibility and $ \ ell_p $ norms中的最短矢量问题
Parameterized Inapproximability of the Minimum Distance Problem over all Fields and the Shortest Vector Problem in all $\ell_p$ Norms
论文作者
论文摘要
我们证明，在任何固定有限的字段上线性代码上的最小距离问题（MDP），并且通过输入距离界定的参数为w [1] - 在任何常数因子内近似。我们还证明了整数晶格上的参数化最短矢量问题（SVP）的结果。具体而言，我们证明$ \ ell_p $ norm中的SVP是W [1] - 在任何固定$ p> 1 $ and w [1] - 在接近$ 2 $ $ 2 $的因子的任何常数因子内近似左右。 （在每种情况下随机减少下显示硬度。） 这些结果回答了Bhattacharyya，Bonnet，Egri，Ghoshal，Karthik C. S.，Lin，Lin，Manurangsi和Marx（ACM，2021年杂志）在参数化MDP和SVP的复杂性上，Bhattacharyya，Egri，Ghoshal，Karthik C. S.，Lin，Manurangsi和Marx（明确提出）的主要问题。对于MDP，他们为二进制线性代码建立了类似的硬度，并使一般字段打开了。对于$ \ ell_p $ norms的高级副总裁，$ p> 1 $，它们在某个恒定因素（取决于$ p $）内表现出无Ximibibibibility的性能，并且在任意恒定的因素上留下了如此硬度。他们还在$ \ ell_1 $ norm中的确切svp的w [1] - hard度上留下了公开。
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.
