Additive approximation for edge-deletion problems

Abstract

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.