稠密3-子超图问题的改进近似算法
Improved approximation algorithm for the Dense-3-Subhypergraph Problem
摘要：
稠密- $3$ -子超图问题的研究始于Chlamt{á}c等人[Approx'16]。输入是全域$U$和$U$的子集的集合${\cal S}$，每个子集具有大小$3$和数字$k$。目标是从全域中选择$k$元素的集合$W$，并最大化集合的数量$S\in {\cal S}$，使得$S\subseteq W$。$U$中的成员称为{ \em顶点}，而${\cal S}$的集合称为{ \em超边}。这是最简单的扩展到超边的情况下，集的大小$2$这是众所周知的密集$k$ -子图的问题。Chlamt{á}c等人提出的稠密$3$ -子超图的最佳比率是$O(n^{0.69783..})$，我们将这个比率改进为$n^{0.61802..}$ .更重要的是，我们给出了一个新的算法，近似密集- $3$ -子超图在一个比率的$\tilde O(n/k)$，这改善了的比率的$O(n^2/k^2)$的Chlamt{á}c等人。我们证明了下的{ \em对数密度猜想}（见Bhaskara等人。[STOC'10]）的比率不能优于$Ω(\sqrt{n})$，并展示了一些情况下，这种最佳可以达到。
英文摘要：
The study of Dense-$3$-Subhypergraph problem was initiated in Chlamt{á}c et al. [Approx'16]. The input is a universe $U$ and collection ${\cal S}$ of subsets of $U$, each of size $3$, and a number $k$. The goal is to choose a set $W$ of $k$ elements from the universe, and maximize the number of sets, $S\in {\cal S}$ so that $S\subseteq W$. The members in $U$ are called {\em vertices} and the sets of ${\cal S}$ are called the {\em hyperedges}. This is the simplest extension into hyperedges of the case of sets of size $2$ which is the well known Dense $k$-subgraph problem. The best known ratio for the Dense-$3$-Subhypergraph is $O(n^{0.69783..})$ by Chlamt{á}c et al. We improve this ratio to $n^{0.61802..}$. More importantly, we give a new algorithm that approximates Dense-$3$-Subhypergraph within a ratio of $\tilde O(n/k)$, which improves the ratio of $O(n^2/k^2)$ of Chlamt{á}c et al. We prove that under the {\em log density conjecture} (see Bhaskara et al. [STOC'10]) the ratio cannot be better than $Ω(\sqrt{n})$ and demonstrate some cases in which this optimum can be attained.
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