Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jonathan Conroy, Arnold Filtser

TL;DR

The paper proves that the shortest-path metric of every $K_r$-minor-free graph admits a padded decomposition with padding parameter $O(\log r)$, resolving a long-standing open question and bridging a large gap from prior bounds. The authors achieve this by developing improved sparse covers for minor-free graphs via a buffered cop decomposition with separator-based refinements, and by establishing a robust reduction from sparse covers to padded decompositions. This combination yields near-optimal, scalable decomposability results with broad algorithmic consequences, including tighter bounds for multiflow/min-cut, flow sparsification, sparse partitions, and metric embeddings in minor-free settings. The work clarifies the structural underpinnings of minor-free graphs and provides a practical framework for designing divide-and-conquer and embedding algorithms with $O(\log r)$ dependence on the minor size $r$.

Abstract

Roughly, a metric space has padding parameter $β$ if for every $Δ>0$, there is a stochastic decomposition of the metric points into clusters of diameter at most $Δ$ such that every ball of radius $γΔ$ is contained in a single cluster with probability at least $e^{-γβ}$. 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.

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

TL;DR

The paper proves that the shortest-path metric of every -minor-free graph admits a padded decomposition with padding parameter , resolving a long-standing open question and bridging a large gap from prior bounds. The authors achieve this by developing improved sparse covers for minor-free graphs via a buffered cop decomposition with separator-based refinements, and by establishing a robust reduction from sparse covers to padded decompositions. This combination yields near-optimal, scalable decomposability results with broad algorithmic consequences, including tighter bounds for multiflow/min-cut, flow sparsification, sparse partitions, and metric embeddings in minor-free settings. The work clarifies the structural underpinnings of minor-free graphs and provides a practical framework for designing divide-and-conquer and embedding algorithms with dependence on the minor size .

Abstract

Roughly, a metric space has padding parameter if for every , there is a stochastic decomposition of the metric points into clusters of diameter at most such that every ball of radius is contained in a single cluster with probability at least . 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 -minor-free graph has padding parameter , 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 -minor-free graphs with improved parameters, and we prove a general reduction from sparse covers to padded decompositions.

Paper Structure

This paper contains 12 sections, 9 theorems, 12 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Sparse Cover
  3. Related Work and Additional Background
  4. Technical Ideas
  5. Preliminaries
  6. Buffered cop decomposition.
  7. From buffered cop decomposition to sparse cover
  8. Selecting separator supernodes in a cop decomposition
  9. Sparse cover construction
  10. From sparse covers to padded decompositions
  11. Applications
  12. Concluding Remarks

Key Result

Theorem 2

Every $K_r$-minor-free graph admits an $(O(\log r), \Omega(1))$-padded decomposition scheme.

Figures (6)

  • Figure 1: An animation of constructing supernodes on an (implict) planar graph, following the cop decomposition of AGGNT19. Each supernode $\eta_i$ is constructed in a connected component $\mathrm{dom}(\eta_i)$ of $G\setminus\cup_{j<i}\eta_j$. The skeleton $T_{\eta_i}$ of $\eta_i$ is a shortest path tree in $\mathrm{dom}(\eta_i)$, with at least one vertex neighboring each previously-created supernode adjacent to $\mathrm{dom}(\eta_i)$. In AGGNT19 the supernode $\eta_i$ is a ball around $T_{\eta_i}$ of radius at most $\Delta$. Later, we discuss buffered cop decomposition CCLMST24 which is a similar object (but with a different construction); there, $\eta_i$ is a connected set of vertices at distance $\le \Delta$ around $T_{\eta_i}$.
  • Figure 2: Top: Graph $G$ partitioned into supernodes of a cop decomposition. Bottom: Partition tree $\mathcal{T}_G$ of the cop decomposition. The supernode $\eta_i$ is the child of the most-recently created supernode adjacent to $\mathrm{dom}(\eta_i)$. The set $\operatorname{Bag}(\eta_i)$ contains $\eta_i$ and all previously-created supernodes adjacent to $\mathrm{dom}(\eta_i)$; it is illustrated to the right of each $\eta_i$ in the partition tree. The bags induce a tree decomposition of $G$.
  • Figure 3: A styled depiction of the incorrect sparse cover algorithm on a graph $G$ (left) and its cop decomposition $\mathcal{T}_G$ (right). On the right: Supernode $X$ is highlighted in red. Deleting $X$ from $\mathcal{T}_G$ creates three connected components $\mathcal{T}_1$, $\mathcal{T}_2$, and $\mathcal{T}_3$. On the left: The supernodes $\{ X, X' \} = \operatorname{Bag}(X)$ form a separator in $G$. The subgraphs induced by $\mathcal{T}_1$, $\mathcal{T}_2$, and $\mathcal{T}_3$ are highlighted in $G$. Red and brown dashed lines outline a cluster grown around $X$ in $\mathrm{dom}(X)$ and a cluster grown around $X'$ in $\mathrm{dom}(X')$. The ball $B$ is not contained in either of the clusters, nor is it contained in a subgraph that is dealt with recursively.
  • Figure 4: A stylized depiction of the divide-and-conquer algorithm for sparse cover. Initially all vertices are active. After deleting supernode $X$ from $\mathcal{T}$, we recuse on each connected component $\mathcal{T}_i$ of $\mathcal{T} \setminus X$ (shown right) and its associated active vertex set $A_i$ (shown left).
  • Figure 5: Illustration of the partition tree $\mathcal{T}_G$ (of the buffered cop decomposition from \ref{['fig:CopDecomp']}), and the output of $\textsc{SeparatorSupernodes}(\mathcal{T}_G)$. Supernodes in $\mathcal{X}$ are red; supernodes not in $\mathcal{X}$ are gray. For each supernode $\eta$ that is not in $\mathcal{X}$, we highlight (in pink) each supernode in $\operatorname{Bag}(\eta)$ that appears in the bag of some ancestor supernode in $\mathcal{X}$; these supernodes witness the fact that $\eta$ is "marked" and does not join $\mathcal{X}$.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Definition 1: Padded Decomposition
  • Theorem 2
  • Definition 3: Sparse Cover
  • Theorem 4: From sparse covers to padded decompositions
  • Theorem 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Lemma 9: Theorem 3.15 of CCLMST24
  • Theorem 11
  • ...and 18 more