Approximation and kernelization for chordal vertex deletion

Abstract

\u3cp\u3eThe Chordal Vertex Deletion (ChVD) problem asks to delete a minimum number of vertices from an input graph to obtain a chordal graph. In this paper we develop a polynomial kernel for ChVD under the parameterization by the solution size. Using a new Erdos-Posa-type packing/covering duality for holes in nearly chordal graphs, we present a polynomial-time algorithm that reduces any instance (G, k) of ChVD to an equivalent instance with poly(k) vertices. The existence of a polynomial kernel answers an open problem posed by Marx in 2006 [D. Marx, ``Chordal Deletion Is Fixed-Parameter Tractable, in Graph-Theoretic Concepts in Computer Science, Lecture Notes in Comput. Sci. 4271, Springer, 2006, pp. 37--48]. To obtain the kernelization, we develop the first poly(opt)-approximation algorithm for ChVD, which is of independent interest. In polynomial time, it either decides that G has no chordal deletion set of size k, or outputs a solution of size \scrO (k\u3csup\u3e4\u3c/sup\u3e log\u3csup\u3e2\u3c/sup\u3e k).\u3c/p\u3