A simple (2+ϵ)-approximation algorithm for Split Vertex Deletion - Drescher, Matthew; Fiorini, Samuel; Huynh, Tony (European journal of combinatorics, 2024)
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 if 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 (Lokshtanov et al., 2020) 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.