发表机构
Rutgers University; Massachusetts Institute of Technology（罗格斯大学; 麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
通过两个改进的多层PCP构造，分别实现拟线性大小和2对2约束，证明了超图顶点覆盖的紧NP难度及指数时间假说下的下界。
AI 中文摘要
我们提出了两个多层PCP的基本构造，在两个方面优于先前的构造。具体来说，我们给出了一个拟线性大小的构造，以及另一个具有2对2约束的构造。利用这些构造，我们针对超图顶点覆盖问题获得了以下结果：\n• 对于k=3，对所有ε>0，在因子1+√2−ε内逼近给定3-均匀超图的最小顶点覆盖是NP难的。此前，[Dinur, Guruswami, Khot, Regev, SICOMP 2005]所知的最近结果达到了2−ε的因子。\n• 对于k≥4，对所有ε>0，在因子k−ε内逼近给定k-均匀超图的最小顶点覆盖是NP难的，这是紧的。先前的工作在假设唯一游戏猜想[Khot, Regev, JCSS 2008]下建立了这一结果，而对于标准NP难度，则得到了较弱的因子k−1−ε[Dinur, Guruswami, Khot, Regev, SICOMP 2005]。\n• 假设指数时间假说，对所有k≥3和ε>0，存在C>0，使得没有2^{n/log^C n}时间的算法能在因子k−1−ε内逼近具有n个顶点的k-均匀超图的最小顶点覆盖。\n证明是使用ChatGPT 5.6 Pro获得的，随后由沟通者重写。
英文摘要
We present two elementary constructions of multilayered PCPs that improve upon prior constructions in two ways. Specifically, we give one construction of quasi-linear size, and another one with $2$-to-$2$ constraints. Using these constructions we obtain the following results for the hypergraph vertex cover problem: $\bullet$ For $k=3$, for all $\varepsilon>0$, approximating the minimum vertex cover of a given $3$-uniform hypergraph within factor $1+\sqrt{2}-\varepsilon$ is NP-hard. Previously, the best known result due to [Dinur, Guruswami, Khot, Regev, SICOMP 2005] achieved a factor of $2-\varepsilon$. $\bullet$ For $k\geq 4$, for all $\varepsilon>0$, approximating the minimum vertex cover of a given $k$-uniform hypergraph within factor $k-\varepsilon$ is NP-hard, which is tight. Previous works established this result assuming the Unique-Games Conjecture [Khot, Regev, JCSS 2008], and a weaker factor of $k-1-\varepsilon$ for standard NP-hardness [Dinur, Guruswami, Khot, Regev, SICOMP 2005]. $\bullet$ Assuming the Exponential Time Hypothesis, for all $k\geq 3$ and $\varepsilon>0$ there is $C>0$ such that no $2^{n/\log^C n}$-time algorithm approximates the minimum vertex cover in a $k$-uniform, $n$-vertex hypergraph within factor $k-1-\varepsilon$. The proofs were obtained using ChatGPT 5.6 Pro and subsequently rewritten by the communicators.
