  A graph property is monotone if it is closed under removal of vertices and
edges. In this paper we consider the following edge-deletion problem; given a
monotone property P and a graph G, compute the smallest number of edge
deletions that are needed in order to turn G into a graph satisfying P. We
denote this quantity by E_P(G).
  Our first result states that for any monotone graph property P, any \epsilon
>0 and n-vertex input graph G one can approximate E_P(G) up to an additive
error of \epsilon n^2
  Our second main result shows that such approximation is essentially best
possible and for most properties, it is NP-hard to approximate E_P(G) up to an
additive error of n^{2-\delta}, for any fixed positive \delta.
  The proof requires several new combinatorial 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_P(G) precisely for dense
monotone properties is NP-hard. We thus answer (in a strong form) a question of
Yannakakis raised in 1981.
