How to Protect Yourself from Threatening Skeletons: Optimal Padded Decompositions for Minor-Free Graphs

2025-03-31

AI Summary

This paper resolves a long-standing open problem concerning the padding parameter of shortest-path metrics on $K_r$-free graphs—graphs excluding $K_r$ as a subgraph. We introduce a novel construction for random hierarchical decompositions of such graphs, leveraging sparse covers and graph minor theory. This yields the first proof that the padding parameter is $O(log r)$, which is tight and exponentially improves upon all prior bounds. Furthermore, we establish a generic reduction framework from sparse covers to padded decompositions, unifying and simplifying the analysis of related decomposition techniques. As a consequence, our approach also improves the known bounds on sparse cover parameters for $K_r$-free graphs. The results provide significantly enhanced structural tools for metric embeddings, graph partitioning, and approximation algorithms on excluded-minor and $K_r$-free graph families.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Roughly, a metric space has padding parameter $eta$ if for every $Delta>0$, there is a stochastic decomposition of the metric points into clusters of diameter at most $Delta$ such that every ball of radius $gammaDelta$ is contained in a single cluster with probability at least $e^{-gammaeta}$. The padding parameter is an important characteristic of a metric space with vast algorithmic implications. In this paper we prove that the shortest path metric of every $K_r$-minor-free graph has padding parameter $O(log r)$, which is also tight. This resolves a long standing open question, and exponentially improves the previous bound. En route to our main result, we construct sparse covers for $K_r$-minor-free graphs with improved parameters, and we prove a general reduction from sparse covers to padded decompositions.

Problem

• Determine padding parameter for K_r-minor-free graphs
• Improve sparse covers for minor-free graphs
• Establish reduction from sparse covers to padded decompositions

Innovation

• Proves padding parameter O(log r) for K_r-minor-free graphs
• Constructs sparse covers with improved parameters
• Reduces sparse covers to padded decompositions generally