On the Lossy Kernelization for Connected Treedepth Deletion Set

Abstract

AbstractWe study the Connected \(\eta \) -Treedepth Deletion problem, where the input instance is an undirected graph G, and an integer k and the objective is to decide whether there is a vertex set \(S \subseteq V(G)\) such that \(|S| \le k\), every connected component of \(G - S\) has treedepth at most \(\eta \) and G[S] is a connected graph. As this problem naturally generalizes the well-studied Connected Vertex Cover problem, when parameterized by the solution size k, Connected \(\eta \) -Treedepth Deletion is known to not admit a polynomial kernel unless \(\mathsf{NP \subseteq coNP/poly}\). This motivates the question of designing approximate polynomial kernels for this problem.In this paper, we show that for every fixed \(0 < \varepsilon \le 1\), Connected \(\eta \) -Treedepth Deletion admits a time-efficient \((1+\varepsilon )\)-approximate kernel of size \(k^{2^{{\mathcal O}(\eta + 1/\varepsilon )}}\) (i.e., a Polynomial-size Approximate Kernelization Scheme).KeywordsTreedepthKernelizationConnected Treedepth Deletion SetLossy Kernelization