### Abstract

A graph property is *monotone* if it is closed under removal of vertices and edges. In this paper we consider the following algorithmic problem, called the edge-deletion problem; given a monotone property $\mathcal{P}$ and a graph $G$, compute the smallest number of edge deletions that are needed in order to turn $G$ into a graph satisfying $\mathcal{P}$. We denote this quantity by $E’_{\mathcal{P}}(G)$. The first result of this paper states that the edge-deletion problem can be efficiently approximated for any monotone property.

- For any fixed $\varepsilon > 0$ and any monotone property $\mathcal{ P}$, there is a deterministic algorithm which, given a graph $G=(V,E)$ of size $n$, approximates $E’_{\mathcal{P}}(G)$ in linear time $O(|V|+|E|)$ to within an additive error of $\varepsilon n^2$.

Given the above, a natural question is for which monotone properties one can obtain better additive approximations of $E’_{\mathcal{P}}$. Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist.

- (1) If there is a bipartite graph that does not satisfy $\mathcal{P}$, then there is a $\delta > 0$ for which it is possible to approximate $E’_{\mathcal{P}}$ to within an additive error of $n^{2-\delta}$ in polynomial time.
- (2) On the other hand, if all bipartite graphs satisfy $\mathcal{P}$, then for any $\delta > 0$ it is NP-hard to approximate $E’_{\mathcal{ P}}$ to within an additive error of $n^{2-\delta}$.

While the proof of (1) is relatively simple, the proof of (2) requires several new ideas and involves tools from Extremal Graph Theory together with spectral techniques. Interestingly, prior to this work it was not even known that computing $E’_{\mathcal{P}}$ *precisely* for the properties in (2) is NP-hard. We thus answer (in a strong form) a question of Yannakakis, who asked in 1981 if it is possible to find a large and natural family of graph properties for which computing $E’_{\mathcal{P}}$ is NP-hard.