On the Lossy Kernelization for Connected Treedepth Deletion Set

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 subset of V(G) such that vertical bar S vertical bar <= 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 NP subset of coNP/poly. This motivates the question of designing approximate polynomial kernels for this problem. In this paper, we show that for every fixed 0 < epsilon <= 1, CONNECTED eta-TREEDEPTH DELETION admits a time-efficient (1+epsilon) -approximate kernel of size k(2O(eta+1/epsilon)) (i.e., a Polynomial-size Approximate Kernelization Scheme).