Optimal Padded Decomposition For Bounded Treewidth Graphs

TL;DR

This paper shows that graphs with bounded treewidth $\mathrm{tw}$ admit a padding decomposition with parameter $O(\log \mathrm{tw})$, substantially improving the dependence on treewidth for a broad class of problems. Central to the approach are tree-ordered nets and a novel tree-partition framework, enabling efficient construction of padded decompositions and sparse covers even when graphs do not have sparse nets. The results yield exponential improvements in flow-cut gaps, 0-extension approximations, and metric embeddings, among others, and imply stronger bounds for flow sparsifiers and universal network problems. By connecting treewidth-based decompositions to padded partition covers, the work advances toward resolving long-standing conjectures for minor-free graphs and provides a flexible toolkit for leveraging small treewidth in divide-and-conquer graph algorithms.

Abstract

A $(β,δ,Δ)$-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 $v\in V$, the probability that $\rm{ball}_G(v,γΔ)$ is entirely contained in the cluster containing $v$ is at least $e^{-βγ}$ for every $γ\in [0,δ]$. 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, $β= Θ(\log n)$. Klein, Plotkin, and Rao showed that $K_r$-minor-free graphs have padding parameter $β= O(r^3)$, which is a significant improvement over general graphs when $r$ is a constant. A long-standing conjecture is to construct a padded decomposition for $K_r$-minor-free graphs with padding parameter $β= O(\log r)$. 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 $\rm{tw}$ admit a padded decomposition with padding parameter $O(\log \rm{tw})$, which is tight. As corollaries, we obtain an exponential improvement in dependency on treewidth in a host of algorithmic applications: $O(\sqrt{ \log n \cdot \log(\rm{tw})})$ flow-cut gap, max flow-min multicut ratio of $O(\log(\rm{tw}))$, an $O(\log(\rm{tw}))$ approximation for the 0-extension problem, an $\ell^{O(\log n)}_\infty$ embedding with distortion $O(\log \rm{tw})$, and an $O(\log \rm{tw})$ bound for integrality gap for the uniform sparsest cut.

Figures (4)

• Figure 4: Illustration of a tree order.
• Figure 5: Converting a tree decomposition in (a) into a tree partition in (b). The thick red edges are the original edges in $G$, and the thin blue edges are the added edges with weight zero.
• Figure 6: Illustrating two rounds of \ref{['alg:CoreFinding']}. (b) In the first round, the algorithm creates (blue) cores from $\mathcal{T}$; the attachment of every bag is $\emptyset$. (c) After the first round, uncovered bags of $\mathcal{T}$ form a forest; these are the white bags. Some bags now have non-empty attachments, illustrated by the gray-shaded regions. (d) In the second round, the algorithm considers each connected component $\mathcal{T}'$ of uncovered bags and creates cores by carving balls $\mathbf{B}_H(\mathtt{Uncov}(B),\Delta)$ in graph $H$ from uncovered vertices of $B$. The attachment allows a core of the 2nd round to grow within a core of the 1st round. (e) The remaining uncovered bags and their attachments after round 2. A core in round 2 carves out a portion of the attachment of $X$.
• Figure 7: (a) Core $R_2$ and graph $G_{\mathcal{T}}[R_2]$ rooted at its center bag, (b) the shadow domain, shadow and strict shadow of core $R_2$.

Theorems & Definitions (38)

• conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• definition 1.4: Padded Partition Cover Scheme
• Theorem 1.5
• definition 1.6
• definition 1.7: Tree Partition
• Lemma 1.7
• definition 2.1: Padded Decomposition
• Lemma 2.2