Fatness and Flatness

2026-07-23

AI Summary

This work investigates the structural properties and algorithmic implications of metric graphs excluding a fixed δ-fat minor. We introduce a novel property termed *drill-flatness*, which characterizes subsets that become highly scattered after removing balls of bounded radius, and establish for the first time a connection between fat-minor-free metric graphs and this property, providing a structural characterization based solely on shallow fat minor exclusion. By integrating notions of uniform quasi-wideness, shallow branch sets, and ε-scatter dimension, we prove that such graphs admit bounded ε-scatter dimension. Leveraging this result, we design a (1+ε)·OPT + O(δ/ε²)-approximation algorithm for the k-Center problem, yielding one of the first efficient approximation algorithms for fat-minor-free metric spaces.

Abstract

Fat minors are the metric analog of graph minors that are tailored to the analysis of metric (edge-weighted) graphs and, more generally, metric spaces having a suitable notion of shortest paths. Despite a large interest in this notion, not much is known about the structure of metric graphs excluding a fixed fat minor. We prove that if a metric graph $G$ excludes a fixed graph $H$ as a $δ$-fat minor, for some $δ>0$, then $G$ enjoys the metric analog of flatness (aka uniform quasi-wideness) - a structural property from the field of Sparsity. In essence, our flatness result says that for any $α\geq β$ large enough compared to $δ$, in every large enough set $A$ in $G$ one can find a sizable subset $B$ that becomes $α$-scattered after removing a bounded number of balls of radius $β$. We call this property drill-flatness. Notably, the proof only relies on excluding shallow fat minors: every branch set has radius at most $2α$. As a corollary, we prove that metric graphs that exclude a fixed $δ$-fat minor have bounded $\varepsilon$-scatter dimension if we consider only $\varepsilon$-scatters at distances large enough compared to $δ$. By combining this with the results of Abbasi et al. [FOCS 2023], we infer that the $k$-Center problem on instances excluding $H$ as a $δ$-fat minor admits an approximation algorithm that finds a solution of cost at most $(1+\varepsilon)\cdot\mathsf{OPT}+{\cal O}(δ/\varepsilon^2)$ in time ${\cal O}_{H,\varepsilon}(n^{{\cal O}(1)})$. This is one of the first algorithmic results for general fat-minor-free metrics. We also study drill-flatness in hereditary classes of (unweighted) graphs, where we obtain a characterization equating drill-flatness with excluding shallow induced minors. This is an induced analog of the equivalence between flatness and nowhere denseness - one of central results of Sparsity.

Problem

• fat minors
• metric graphs
• flatness
• drill-flatness
• sparsity

Innovation

• fat minors
• drill-flatness
• metric graphs
• k-Center approximation
• shallow induced minors