Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter
Résumé
Kernelization is a concept that enables the formal mathematical analysis of data reduction through the framework of parameterized complexity. Intensive research into the \vertexcover problem has shown that there is a preprocessing algorithm which given an instance $(G,k)$ of \vertexcover outputs an equivalent instance $(G',k')$ in polynomial time with the guarantee that $G'$ has at most $2k'$ vertices (and thus $O((k')^2)$ edges) with $k' \leq k$. Using the terminology of parameterized complexity we say that \kvertexcover has a kernel with $2k$ vertices. There is complexity-theoretic evidence that both $2k$ vertices and $\Theta(k^2)$ edges are optimal for the kernel size. In this paper we consider the \vertexcover problem with a different parameter, the size $\fvs(G)$ of a minimum feedback vertex set for $G$. This refined parameter is structurally smaller than the parameter $k$ associated to the vertex covering number $\vc(G)$ since $\fvs(G) \leq \vc(G)$ and the difference can be arbitrarily large. We give a kernel for \vertexcover with a number of vertices that is cubic in $\fvs(G)$: an instance $(G,X,k)$ of \vertexcover, where $X$ is a feedback vertex set for $G$, can be transformed in polynomial time into an equivalent instance $(G',X',k')$ such that $k' \leq k$, $|X'| \leq |X|$ and most importantly $|V(G')| \leq 2k$ and $|V(G')| \in O(|X'|^3)$. A similar result holds when the feedback vertex set $X$ is not given along with the input. In sharp contrast we show that the \wvertexcover problem does not have a polynomial kernel when parameterized by $\fvs(G)$ unless the polynomial hierarchy collapses to the third level (\collapse). Our work is one of the first examples of research in kernelization using a non-standard parameter, and shows that this approach can yield interesting computational insights. To obtain our results we make extensive use of the combinatorial structure of independent sets in forests.
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