arXiv 1905.065032019-05-18cs.CC

精确覆盖与最近码字的参数化不可逼近性

Parameterized Inapproximability of Exact Cover and Nearest Codeword

Venkatesan Guruswami, Patrick Lin

备注：: An error was discovered in the proof of Lemma 3.3

摘要：

$k$-ExactCover问题是ExactCover问题的参数化版本，其中我们给出了一个论域$U$，一个$U$子集的集合$S$，以及一个整数$k$，任务是确定$U$是否可以划分为$S$中的$k$集合。这是集合覆盖问题的一个自然扩展;虽然在参数化的情况下我们知道它在精确的情况下是$W[1]$ -完全的，但它的参数化的复杂性关于可近似性还没有得到很好的理解。我们证明，假设ETH，对于一些$γ> 0$有没有时间$f(k) \cdot N^{γk}$算法，可以，给定一个$k$ -ExactCover实例$I$，区分的情况下，$I$有一个确切的覆盖的大小$k$和的情况下，每个集合覆盖的$I$的大小至少$\frac14 \sqrt[k]{\frac{\log N}{\log \log N}}$。这排除了甚至多于FPT算法，并且还排除了近似比仅取决于参数$k$的任何算法。通过假设SETH，我们反而改进了对于任何$\varepsilon > 0$需要时间$f(k) \cdot N^{k - \varepsilon}$的下限。在这项工作中，我们还扩展了不可逼近的结果$k$ -最近的码字（$k$ -NCP）的问题。具体而言，给定生成矩阵$A \in \mathbb{F}_2^{m \times n}$，矢量$y \in \mathbb{F}_2^m$，和参数$k$，我们表明，它是很难区分的情况下，存在一个码字的距离最多$k$从$y$的情况下，每个码字的距离至少$\frac18 \sqrt[k]{\frac{\log N}{\log \log N}}$从$y$。这改进了最著名的参数化不可近似性结果，该结果排除了因子为$\text{poly} (\log k)$的近似，但要求我们假设ETH而不是$W[1] \neq FPT$。

英文摘要：

The $k$-ExactCover problem is a parameterized version of the ExactCover problem, in which we are given a universe $U$, a collection $S$ of subsets of $U$, and an integer $k$, and the task is to determine whether $U$ can be partitioned into $k$ sets in $S$. This is a natural extension of the well-studied SetCover problem; though in the parameterized regime we know it to be $W[1]$-complete in the exact case, its parameterized complexity with respect to approximability is not well understood. We prove that, assuming ETH, for some $γ> 0$ there is no time $f(k) \cdot N^{γk}$ algorithm that can, given a $k$-ExactCover instance $I$, distinguish between the case where $I$ has an exact cover of size $k$ and the case where every set cover of $I$ has size at least $\frac14 \sqrt[k]{\frac{\log N}{\log \log N}}$. This rules out even more than FPT algorithms, and additionally rules out any algorithm whose approximation ratio depends only on the parameter $k$. By assuming SETH, we instead improve the lower bound to requiring time $f(k) \cdot N^{k - \varepsilon}$, for any $\varepsilon > 0$. In this work we also extend the inapproximability result to the $k$-Nearest-Codeword ($k$-NCP) problem. Specifically, given a generator matrix $A \in \mathbb{F}_2^{m \times n}$, a vector $y \in \mathbb{F}_2^m$, and the parameter $k$, we show that it is hard to distinguish between the case where there exists a codeword with distance at most $k$ from $y$ and the case where every codeword has distance at least $\frac18 \sqrt[k]{\frac{\log N}{\log \log N}}$ from $y$. This improves the best known parameterized inapproximability result, which rules out approximations with a factor of $\text{poly} (\log k)$, but requires us to assume ETH instead of $W[1] \neq FPT$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文