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We study the CONNECTED \eta-TREEDEPTH DELETION problem where the inputinstance is an undireted graph G = (V, E) and an integer k. The objective is todecide if G has a set S \subseteq V(G) of at most k vertices such that G - Shas treedepth at most \eta and G[S] is connected. As this problem naturallygeneralizes the well-known CONNECTED VERTEX COVER, when parameterized bysolution size k, the CONNECTED \eta-TREEDEPTH DELETION does not admitpolynomial kernel unless NP \subseteq coNP/poly. This motivates us to design anapproximate kernel of polynomial size for this problem. In this paper, we showthat for every 0 < \epsilon <= 1, CONNECTED \eta-TREEDEPTH DELETION SET admitsa (1+\epsilon)-approximate kernel with O(k^{2^{\eta + 1/\epsilon}}) vertices,i.e. a polynomial-sized approximate kernelization scheme (PSAKS).

An Improved Time-Efficient Approximate Kernelization for Connected Treedepth Deletion Set