Strong Hardness of Approximation for Tree Transversals

Let $H$ be a fixed graph. The $H$-Transversal problem, given a graph $G$, asks to remove the smallest number of vertices from $G$ so that $G$ does not contain $H$ as a subgraph. While a simple $|V(H)|$-approximation algorithm exists and is believed to be tight for every $2$-vertex-connected $H$, the best hardness of approximation for any tree was $\Omega(\log |V(H)|)$-inapproximability when $H$ is a star. In this paper, we identify a natural parameter $\Delta$ for every tree $T$ and show that $T$-Transversal is NP-hard to approximate within a factor $(\Delta - 1 -\varepsilon)$ for an arbitrarily small constant $\varepsilon > 0$. As a corollary, we prove that there exists a tree $T$ such that $T$-Transversal is NP-hard to approximate within a factor $\Omega(|V(T)|)$, exponentially improving the best known hardness of approximation for tree transversals.