Preprocessing for Treewidth
SIAM Journal on Discrete Mathematics(2013)
摘要
The notion of treewidth plays an important role in theoretical and practical studies of graph problems. It has
been recognized that, especially in practical environments, when computing the treewidth of a graph it is
invaluable to first apply an array of preprocessing rules that simplify and shrink it. This work seeks to
prove rigorous performance guarantees for such preprocessing rules---known rules as well as more recent ones---by
studying them
in the framework of kernelization from parameterized complexity. It is known that the NP-complete problem of
determining whether a given graph $G$ has treewidth at most $k$ admits no polynomial-time preprocessing
algorithm that reduces any input instance to size polynomial in $k$, unless NP $\subseteq$ coNP/poly and the
polynomial hierarchy collapses to its third level. In this paper we therefore consider structural graph
measures larger than treewidth, and determine whether efficient preprocessing can shrink the instance size to
a polynomial in such a parameter value. We prove that, given an instance $(G,k)$ of treewidth, we can
efficiently reduce its size to $\mathcal{O}(\mathrm{\textsc{fvs}}(G)^4)$ vertices, where $\mathrm{\textsc{fvs}}(G)$
is the size of a minimum feedback
vertex set in $G$. We can also prove a size reduction to $\mathcal{O}(\mathrm{\textsc{vc}}(G)^3)$ vertices,
where $\mathrm{\textsc{vc}}(G)$ is the size
of a minimum vertex cover. Phrased in the language of parameterized complexity, we show that Treewidth has a
polynomial kernel when parameterized by the size of a given feedback vertex set, and also by the size of a
vertex cover. In contrast we show that Treewidth parameterized by the vertex-deletion distance to a single
clique and Weighted Treewidth parameterized by the size of a vertex cover do not admit polynomial
kernelizations unless NP $\subseteq$ coNP/poly.
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关键词
parameterized complexity,kernelization,preprocessing,treewidth,05C85,68Q25,68W05
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