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The $H$-Free Edge Deletion problem asks, for a given graph $G$ and an integer$k$, whether it is possible to delete at most $k$ edges from $G$ to make it$H$-free, that is, not containing $H$ as an induced subgraph. The $H$-Free EdgeCompletion problem is defined similarly, but we add edges instead of deletingthem. The study of these two problem families has recently been the subject ofintensive studies from the point of view of parameterized complexity andkernelization. In particular, it was shown that the problems do not admitpolynomial kernels (under plausible complexity assumptions) for almost allgraphs $H$, with several important exceptions occurring when the class of$H$-free graphs exhibits some structural properties.  In this work we complement the parameterized study of edge modificationproblems to $H$-free graphs by considering their approximability. We prove thatwhenever $H$ is $3$-connected and has at least two non-edges, then both$H$-Free Edge Deletion and $H$-Free Edge Completion are very hard toapproximate: they do not admit $\mathrm{poly}(\mathsf{OPT})$-approximation inpolynomial time, unless $\mathrm{P}=\mathrm{NP}$, or even in timesubexponential in $\mathsf{OPT}$, unless the Exponential Time Hypothesis fails.The assumption of the existence of two non-edges appears to be important: weshow that whenever $H$ is a complete graph without one edge, then $H$-Free EdgeDeletion is tightly connected to the Min Horn problem, whose approximability isstill open. Finally, in an attempt to extend our hardness results beyond$3$-connected graphs, we consider the cases of $H$ being a path or a cycle, andwe achieve an almost complete dichotomy there.

Hardness of approximation for H-free edge modification problems