Additive approximation for edge-deletion problems

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). The first result of this paper states that the edge-deletion problem can be efficiently approximated for any monotone property. 1) For any ε > 0 and any monotone property P, there is a deterministic algorithm, which given a graph G of size n, approximates E_P´(G) in time O(n²) to within an additive error of εn². Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E_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 P, then there is a δ > 0 for which it is possible to approximate E_P´ to within an additive error of n^2-δ in polynomial time. 2) On the other hand, if all bipartite graphs satisfy P, then for any δ > 0 it is NP-hard to approximate E_P´ to within an additive error of n^2-δ. While the proof of (1) is simple, the proof of (2) requires several new ideas and involves tools from extremal graph theory together with spectral techniques. This approach may be useful for obtaining other hardness of approximation results. Interestingly, prior to this work it was not even known that computing E_P´ precisely for the properties in (2) is NP-hard. We thus answer (in a strong form) a question of Yannakakis [1981], who asked in 1981 if it is possible to find a large and natural family of graph properties for which computing E_P´ is NP-hard.