Hitting Forbidden Minors: Approximation and Kernelization

Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, Saket Saurabh
2016 SIAM Journal on Discrete Mathematics  
We study a general class of problems called p-F-Deletion problems. In an p-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 p-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,
more » ... here t is a fixed integer. an approximation algorithm achieving an approximation ratio of O(log 3/2 OP T ), where OP T is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when F only 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.
doi:10.1137/140997889 fatcat:ibdw3u7a6vel3hfat3j5rjhjom