Abstract
We 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).
M. S. Ramanujan is supported by Engineering and Physical Sciences Research Council (EPSRC) grants EP/V007793/1 and EP/V044621/1.
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Notes
- 1.
Due to lack of space, omitted proofs or the proofs marked \(\star \) can be found in the full version.
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Eiben, E., Majumdar, D., Ramanujan, M.S. (2022). On the Lossy Kernelization for Connected Treedepth Deletion Set. In: Bekos, M.A., Kaufmann, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2022. Lecture Notes in Computer Science, vol 13453. Springer, Cham. https://doi.org/10.1007/978-3-031-15914-5_15
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