In the $(k,h)$-SetCover problem, we are given a collection $\mathcal{S}$ of sets over a universe $U$, and the goal is to distinguish between the case that $\mathcal{S}$ contains $k$ sets which cover $U$, from the case that at least $h$ sets in $\mathcal{S}$ are needed to cover $U$. Lin (ICALP'19) recently showed a gap creating reduction from the $(k,k+1)$-SetCover problem on universe of size $O_k(\log |\mathcal{S}|)$ to the $\left(k,\sqrt[k]{\frac{\log|\mathcal{S}|}{\log\log |\mathcal{S}|}}\cdot k\right)$-SetCover problem on universe of size $|\mathcal{S}|$. In this paper, we prove a more scalable version of his result: given any error correcting code $C$ over alphabet $[q]$, rate $\rho$, and relative distance $\delta$, we use $C$ to create a reduction from the $(k,k+1)$-SetCover problem on universe $U$ to the $\left(k,\sqrt[2k]{\frac{2}{1-\delta}}\right)$-SetCover problem on universe of size $\frac{\log|\mathcal{S}|}{\rho}\cdot|U|^{q^k}$. Lin established his result by composing the input SetCover instance (that has no gap) with a special threshold graph constructed from extremal combinatorial object called universal sets, resulting in a final SetCover instance with gap. Our reduction follows along the exact same lines, except that we generate the threshold graphs specified by Lin simply using the basic properties of the error correcting code $C$. We use the same threshold graphs mentioned above to prove inapproximability results, under W[1]$\neq$FPT and ETH, for the $k$-MaxCover problem introduced by Chalermsook et al. (SICOMP'20). Our inapproximaiblity results match the bounds obtained by Karthik et al. (JACM'19), although their proof framework is very different, and involves generalization of the distributed PCP framework. Prior to this work, it was not clear how to adopt the proof strategy of Lin to prove inapproximability results for $k$-MaxCover.

中文翻译：

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关于参数化集合覆盖和标签覆盖逼近的硬度：来自纠错码的阈值图

在 $(k,h)$-SetCover 问题中，我们给定了一个集合 $\mathcal{S}$ 在一个宇宙 $U$ 上的集合，目标是区分 $\mathcal{S} $ 包含覆盖 $U$ 的 $k$ 集，从需要在 $\mathcal{S}$ 中设置的至少 $h$ 集来覆盖 $U$ 的情况来看。Lin (ICALP'19) 最近展示了从 $(k,k+1)$-SetCover 问题在大小为 $O_k(\log |\mathcal{S}|)$ 到 $\left( k,\sqrt[k]{\frac{\log|\mathcal{S}|}{\log\log |\mathcal{S}|}}\cdot k\right)$-SetCover 问题在大小为 $ 的宇宙上|\mathcal{S}|$。在本文中，我们证明了他的结果的更具可扩展性的版本：给定字母表 $[q]$ 上的任何纠错码 $C$、比率 $\rho$ 和相对距离 $\delta$，我们使用 $C$创建从宇宙 $U$ 上的 $(k,k+1)$-SetCover 问题到 $\left(k, \sqrt[2k]{\frac{2}{1-\delta}}\right)$-SetCover 问题在大小为 $\frac{\log|\mathcal{S}|}{\rho}\cdot| 的宇宙上U|^{q^k}$。Lin 通过将输入 SetCover 实例（没有间隙）与由称为通用集的极值组合对象构造的特殊阈值图组合来确定他的结果，从而产生具有间隙的最终 SetCover 实例。我们的减少遵循完全相同的路线，除了我们仅使用纠错码 $C$ 的基本属性生成 Lin 指定的阈值图。对于 Chalermsook 等人提出的 $k$-MaxCover 问题，我们使用上面提到的相同阈值图来证明在 W[1]$\neq$FPT 和 ETH 下的不可近似性结果。(SICOMP'20)。我们的不可近似性结果与 Karthik 等人获得的界限相匹配。(JACM'19), 尽管他们的证明框架非常不同，并且涉及分布式PCP框架的泛化。在这项工作之前，不清楚如何采用 Lin 的证明策略来证明 $k$-MaxCover 的不可近似性结果。