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Additive Approximation for Edge-Deletion Problems Asaf Shapira

Summary: Additive Approximation for Edge-Deletion Problems
Noga Alon
Asaf Shapira
Benny Sudakov
A graph property is monotone if it is closed under removal of vertices and edges. In this
paper we consider the following algorithmic problem, called the 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 EP (G). The
first result of this paper states that the edge-deletion problem can be efficiently approximated for
any monotone property.
For any fixed > 0 and any monotone property P, there is a deterministic algorithm, which
given a graph G = (V, E) of size n, approximates EP (G) in linear time O(|V | + |E|) to
within an additive error of n2
Given the above, a natural question is for which monotone properties one can obtain better
additive approximations of EP . 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 > 0 for which it is
possible to approximate EP to within an additive error of n2-


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics