Abstract
Using the framework of kernelization we study whether efficient preprocessing schemes for the Treewidth problem can give provable bounds on the size of the processed instances. Assuming the AND-distillation conjecture to hold, the standard parameterization of Treewidth does not have a kernel of polynomial size and thus instances (G,k) of the decision problem of Treewidth cannot be efficiently reduced to equivalent instances of size polynomial in k. In this paper, we consider different parameterizations of Treewidth. We show that Treewidth has a kernel with \(\mathcal{O}(\ell^3)\) vertices, where ℓ denotes the size of a vertex cover, and a kernel with \(\mathcal{O}(\ell^4)\) vertices, where ℓ denotes the size of a feedback vertex set. This implies that given an instance (G,k) of Treewidth we can efficiently reduce its size to \(\mathcal{O}((\ell^*)^4)\) vertices, where ℓ* is the size of a minimum feedback vertex set in G. In contrast, we show that Treewidth parameterized by the vertex-deletion distance to a co-cluster graph and Weighted Treewidth parameterized by the size of a vertex cover do not have polynomial kernels unless NP ⊆ coNP/poly. Treewidth parameterized by the target value plus the deletion distance to a cluster graph has no polynomial kernel unless the AND-distillation conjecture does not hold.KeywordsVertex CoverPolynomial KernelTree DecompositionCommon NeighborChordal GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.