Degree-3 Treewidth Sparsifiers

We study treewidth sparsifiers. Informally, given a graph G of treewidth k, a treewidth sparsifier H is a minor of G, whose treewidth is close to k, |V (H)| is small, and the maximum vertex degree in H is bounded. Treewidth sparsifiers of degree 3 are of particular interest, as routing on node-disjoint paths, and computing minors seems easier in sub-cubic graphs than in general graphs. In this paper we describe an algorithm that, given a graph G of treewidth k, computes a topological minor H of G such that (i) the treewidth of H is Ω(k/polylog(k)); (ii) |V (H)| = O(k); and (iii) the maximum vertex degree in H is 3. The running time of the algorithm is polynomial in |V (G)| and k. Our result is in contrast to the known fact that unless NP ⊆ coNP/poly, treewidth does not admit polynomial-size kernels. One of our key technical tools, which is of independent interest, is a construction of a small minor that preserves node-disjoint routability between two pairs of vertex subsets. This is closely related to the open question of computing small goodquality vertex-cut sparsifiers that are also minors of the original graph.