Additive Approximation of Generalized Turán Questions

For graphs G and T, and a family of graphs F let ex (G, T, F) denote the maximum possible number of copies of T in an F-free subgraph of G. We investigate the algorithmic aspects of calculating and estimating this function. We show that for every graph T, finite family F and constant ϵ> 0 there is a polynomial time algorithm that approximates ex (G, T, F) for an input graph G on n vertices up to an additive error of ϵn^v(T). We also consider the possibility of a better approximation, proving several positive and negative results, and suggesting a conjecture on the exact relation between T and F for which no significantly better approximation can be found in polynomial time unless P= NP.