Optimal Padded Decomposition For Bounded Treewidth Graphs

A (β, δ, A)-padded decomposition of an edge-weighted graph G = ( V, E, w) is a stochastic decomposition into clusters of diameter at most Δ such that for every vertex v ∈ V, the probability that B_G(v, γΔ) is entirely contained in the cluster containing v is at least e^-βγ for every γ ∈ [0, δ]. Padded decompositions have been studied for decades and have found numerous applications, including metric embedding, multicommodity flow-cut gap, multicut, and zero extension problems, to name a few. In these applications, parameter β, called the padding parameter, is the most important parameter since it decides either the distortion or the approximation ratios. For general graphs with n vertices, the padding parameter β is known to be in ⊖(log n). Klein, Plotkin, and Rao [52] (KPR) showed that Kr-minor-free graphs have padding parameter β = O(r³), which is a significant improvement over general graphs when r is a constant However, when r = Q(log n), the padding parameter in KPR decomposition can be much worse than logn. A long-standing conjecture is that constructing a padded decomposition for Kr-minor-free graphs is possible with padding parameter β = O(logr). Despite decades of research, the best-known result is β = O(r), even for graphs with treewidth at most r. In this work, we make significant progress toward the aforementioned conjecture by showing that graphs with treewidth tw admit a padded decomposition with padding parameter O(logtw), which is tight Our padding parameter is strictly better than O(logn) whenever tw = no(1), and is never worse than what is known for general graphs. As corollaries, we obtain an exponential improvement in dependency on treewidth in a host of algorithmic applications: flow-cut gap, the maxflow-min multicut ratio of O(log(tw)), an O(log(tw)) approximation for the O-extension problem, an embedding with distortion O(logtw), and an O(logtw) bound for integrality gap for the uniform sparsest cut.