胖度与平坦度
Fatness and Flatness
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中文总结 AI 辅助
研究排除固定胖子图的度量图结构，证明其具有钻孔平坦度性质，结合其他结果得出\(k -\)中心问题的近似算法，还研究了图遗传类中的钻孔平坦度并给出特征描述。
中文摘要 AI 辅助
胖子图是图子图的度量类似物，适用于度量（边加权）图以及更一般地具有合适最短路径概念的度量空间的分析。尽管对该概念有很大兴趣，但对于排除固定胖子图的度量图结构知之甚少。我们证明，如果度量图\(G\)排除固定图\(H\)作为\(\delta -\)胖子图，对于某个\(\delta>0\)，那么\(G\)具有平坦度（又名均匀拟宽性）的度量类似物——稀疏性领域的一种结构性质。本质上，我们的平坦度结果表明，对于与\(\delta\)相比足够大的任何\(\alpha\geq\beta\)，在\(G\)中的每个足够大的集合\(A\)中，可以找到一个相当大的子集\(B\)，在移除有限数量半径为\(\beta\)的球后变得\(\alpha -\)分散。我们称此性质为钻孔平坦度。值得注意的是，证明仅依赖于排除浅胖子图：每个分支集的半径至多为\(2\alpha\)。作为推论，我们证明，如果仅考虑与\(\delta\)相比足够大距离处的\(\varepsilon -\)分散，排除固定\(\delta -\)胖子图的度量图具有有界的\(\varepsilon -\)分散维数。通过将此与Abbasi等人[FOCS 2023]的结果相结合，我们推断在排除\(H\)作为\(\delta -\)胖子图的实例上的\(k -\)中心问题允许一种近似算法，该算法在时间\({\cal O}_{H,\varepsilon}(n^{{\cal O}(1)})\)内找到成本至多为\((1 + \varepsilon)\cdot\mathsf{OPT}+{\cal O}(\delta/\varepsilon^2)\)的解。这是关于一般无胖子图度量的首批算法结果之一。我们还研究了（未加权）图的遗传类中的钻孔平坦度，在其中我们得到了一个将钻孔平坦度与排除浅诱导子图等同起来的特征描述。这是平坦度与无处稠密性等价性的诱导类似物——稀疏性的核心结果之一。
英文摘要
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.
