Optimal Padded Decomposition For Bounded Treewidth Graphs

2024-07-17

arXiv.org

AI Summary

This work resolves a long-standing open problem on lower bounds for the padding parameter $eta$ in padded decompositions of bounded-treewidth graphs. For graphs of treewidth $mathrm{tw}$, we construct the first randomized padded decomposition with padding parameter $eta = O(log mathrm{tw})$, improving the prior best bound of $O(mathrm{tw})$ to the asymptotically tight bound. This confirms the $eta = O(log r)$ conjecture for $K_r$-minor-free graphs in the special case of bounded-treewidth graphs. Our method leverages the tree decomposition structure and integrates randomized hierarchical clustering, weighted distance truncation, and locality-sensitive sampling, supported by probabilistic analysis and combinatorial embedding techniques. The decomposition yields several algorithmic advances: the flow-cut gap drops to $O(sqrt{log n cdot log mathrm{tw}})$; the multicommodity flow-cut ratio, $0$-extension approximation ratio, $ell_infty$-embedding distortion, and sparsest-cut integer gap all improve to $O(log mathrm{tw})$, achieving exponential improvements over prior results.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A $(eta,delta,Delta)$-padded decomposition of an edge-weighted graph $G = (V,E,w)$ is a stochastic decomposition into clusters of diameter at most $Delta$ such that for every vertex $vin V$, the probability that $ m{ball}_G(v,gammaDelta)$ is entirely contained in the cluster containing $v$ is at least $e^{-etagamma}$ for every $gamma in [0,delta]$. 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 $eta$, 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, $eta = Theta(log n)$. Klein, Plotkin, and Rao showed that $K_r$-minor-free graphs have padding parameter $eta = O(r^3)$, 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 $eta = O(log r)$. Despite decades of research, the best-known result is $eta = O(r)$, 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 $ m{tw}$ admit a padded decomposition with padding parameter $O(log m{tw})$, which is tight. As corollaries, we obtain an exponential improvement in dependency on treewidth in a host of algorithmic applications: $O(sqrt{ log n cdot log( m{tw})})$ flow-cut gap, max flow-min multicut ratio of $O(log( m{tw}))$, an $O(log( m{tw}))$ approximation for the 0-extension problem, an $ell^{O(log n)}_infty$ embedding with distortion $O(log m{tw})$, and an $O(log m{tw})$ bound for integrality gap for the uniform sparsest cut.

Problem

• Improving padding parameter for bounded treewidth graphs
• Developing optimal padded decomposition with O(log tw)
• Enhancing algorithmic applications via treewidth-dependent bounds

Innovation

• Padded decomposition with O(log tw) padding parameter
• Tight bound for bounded treewidth graphs
• Improves flow-cut gap and approximation ratios