TREE DELETION SET HAS A POLYNOMIAL KERNEL (BUT NO OPTO(1) APPROXIMATION)

In the Tree Deletion Set problem the input is a graph G together with an integer k. The objective is to determine whether there exists a set S of at most k vertices such that G \ S is a tree. The problem is NP-complete and even NP-hard to approximate within any factor of OPTc for any constant c. In this paper we give an O(k(5)) size kernel for the Tree Deletion Set problem. An appealing feature of our kernelization algorithm is a new reduction rule, based on systems of linear equations, that we use to handle the instances on which TREE DELETION SET is hard to approximate.