A simple (2+ε)-approximation algorithm for Split Vertex Deletion
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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) →ℚ_≥ 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.
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