Hitting Forbidden Minors: Approximation and Kernelization

Abstract

We study a general class of problems called F -Deletion problems. In an F -Deletion problem, we are asked whether a subset of at most k vertices can be deleted from a graph G such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F -Deletion problem when F contains a planar graph. We give• a linear vertex kernel on graphs excluding t-claw K 1,t , the star with t leves, as an induced subgraph, where t is a fixed integer.• an approximation algorithm achieving an approximation ratio of O(log 3/2 OPT ), where OPT is the size of an optimal solution on general undirected graphs.Finally, we obtain polynomial kernels for the case when F contains graph θ c as a minor for a fixed integer c. The graph θ c consists of two vertices connected by c parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes.

Kernelization using structural parameters on sparse graph classes

“…Our algorithm is therefore non-constructive, as are all previous algorithms in the meta-kernelization line of work [1,4,25,26].…”

Quick but Odd Growth of Cacti

“…Since Θ is a connected planar graph we obtain a c k n O(1) time algorithm as a corollary to the main results in [8,11,12]. It also has O(k 2 ) kernel [7]. However, we are not aware of exact value of c as all these algorithms use a protrusion subroutine [2].…”

On a generalization of Nemhauser and Trotter's local optimization theorem

“…To find C ′ and I ′ satisfying Basic Condition, we need to find a special ≤ (d + 1)-star packing from X to Y , which can be computed by the algorithms for finding maximum matchings in bipartite graphs. Note that the idea of computing ≤ (d + 1)-stars from X and Y has been used to solve some other problems in references [25,16,11]. We consider the bipartite graph H = (X, Y, E H ) with edge set E H being the set of edges between X and Y in G, and are going to find a ≤ (d + 1)-star packing from X to Y in H. Note that a Y -vertex no adjacent to any vertex in X will become a degree-0 vertex in H. We construct an auxiliary bipartite graph…”