 
Summary: Hardness of edgemodification problems
Noga Alon
Uri Stav
September 20, 2008
Abstract
For a graph property P consider the following computational problem. Given an input graph
G, what is the minimum number of edge modifications (additions and/or deletions) that one
has to apply to G in order to turn it into a graph that satisfies P? Namely, what is the edit
distance (G, P) of a graph G from satisfying P. Clearly, the computational complexity of
such a problem strongly depends on P. For over 30 years this family of computational problems
has been studied in several contexts and various algorithms, as well as hardness results, were
obtained for specific graph properties.
Alon, Shapira and Sudakov studied in [3] the approximability of the computational problem
for the family of monotone graph properties, namely properties that are closed under removal
of edges and vertices. They describe an efficient algorithm that achieves an o(n2
) additive
approximation to (G, P) for any monotone property P, where G is an nvertex input graph,
and show that the problem of achieving an O(n2
) additive approximation is NPhard for
most monotone proeprties. The methods in [3] also provide a polynomial time approximation
