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 $\delta >0$, SVD does not admit a $(2-\delta)$-approximation algorithm, unless P=NP or the Unique Games Conjecture fails. For every $\epsilon >0$, Lokshtanov, Misra, Panolan, Philip, and Saurabh recently gave a randomized $(2+\epsilon)$-approximation algorithm for SVD. In this work we give an extremely simple deterministic $(2+\epsilon)$-approximation algorithm for SVD.

中文翻译：

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一种用于分割顶点删除的简单 $(2+\epsilon)$-近似算法

分裂图是一种图，其顶点集可以划分为一个团和一个稳定集。给定一个图 $G$ 和权重函数 $w: V(G) \to \mathbb{Q}_{\geq 0}$，拆分顶点删除 (SVD) 问题要求找到顶点的最小权重集 $X $ 使得 $GX$ 是一个分裂图。很容易证明一个图是分裂图，当且仅当它不包含 $4$-cycle、$5$-cycle 或作为诱导子图的两条边匹配。因此，SVD 承认一个简单的 $5$-近似算法。另一方面，对于每一个 $\delta >0$，SVD 不承认 $(2-\delta)$-近似算法，除非 P=NP 或唯一博弈猜想失败。对于每一个 $\epsilon >0$，Lokshtanov、Misra、Panolan、Philip 和 Saurabh 最近给出了一个随机的 $(2+\epsilon)$ 近似算法用于 SVD。