论文标题
一个简单的$（2+ε）$ - 拆分顶点删除的近似算法
A simple $(2+ε)$-approximation algorithm for Split Vertex Deletion
论文作者
论文摘要
拆分图是一个图形，其顶点集可以分为一个集团和稳定的集合。给定图形$ g $和权重函数$ w：v（g）\ to \ mathbb {q} _ {\ geq 0} $，拆分顶点删除（svd）问题要求找到最小重量的顶点$ x $，这样$ g-x $是一个拆分图。很容易证明图形是一个拆分图，只有它不包含$ 4 $循环，$ 5 $循环或作为诱导子图的两边匹配。因此，SVD承认了一个简单的$ 5 $ APPRXIMATION算法。另一方面，对于每$δ> 0 $，SVD不承认$（2-δ）$ - 近似算法，除非p = np或独特的游戏猜想失败。 对于SVD的每一个$ε> 0 $，Lokshtanov，Misra，Panolan，Philip和Saurabh最近提供了一个随机的$（2+ε）$ - 近似算法。在这项工作中，我们为SVD提供了一种非常简单的确定性$（2+ε）$ - 近似算法。
A split graph is a graph whose vertex set can be partitioned into a clique and a stable set. Given a graph $G$ and weight function $w: V(G) \to \mathbb{Q}_{\geq 0}$, the Split Vertex Deletion (SVD) problem asks to find a minimum weight set of vertices $X$ such that $G-X$ is a split graph. It is easy to show that a graph is a split graph if and only it it does not contain a $4$-cycle, $5$-cycle, or a two edge matching as an induced subgraph. Therefore, SVD admits an easy $5$-approximation algorithm. On the other hand, for every $δ>0$, SVD does not admit a $(2-δ)$-approximation algorithm, unless P=NP or the Unique Games Conjecture fails. For every $ε>0$, Lokshtanov, Misra, Panolan, Philip, and Saurabh recently gave a randomized $(2+ε)$-approximation algorithm for SVD. In this work we give an extremely simple deterministic $(2+ε)$-approximation algorithm for SVD.
