Preprocessing for Treewidth: A Combinatorial Analysis through Kernelization

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 ANDdistillation 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 O(`) vertices, where ` denotes the size of a vertex cover, and a kernel with O(`) vertices, where ` 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 O((`∗)4) vertices, where `∗ 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.