A graph property is monotone if it is closed under removal of vertices and edges. In this talk 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. Given the above, a natural question is for which monotone properties can one obtain better additive approximations of E_P(G). Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist. We will show that for any dense monotone property, that is, a property for which there are graphs on n vertices with Omega (n^2) edges that satisfy it, 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 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. (Joint work with N. Alon and A. Shapira.
