The notion of shortcut partition, introduced recently by Chang, Conroy, Le, Milenkovi\'c, Solomon, and Than [CCLMST23], is a new type of graph partition into low-diameter clusters. Roughly speaking, the shortcut partition guarantees that for every two vertices $u$ and $v$ in the graph, there exists a path between $u$ and $v$ that intersects only a few clusters. They proved that any planar graph admits a shortcut partition and gave several applications, including a construction of tree cover for arbitrary planar graphs with stretch $1+\varepsilon$ and $O(1)$ many trees for any fixed $\varepsilon \in (0,1)$. However, the construction heavily exploits planarity in multiple steps, and is thus inherently limited to planar graphs. In this work, we breach the "planarity barrier" to construct a shortcut partition for $K_r$-minor-free graphs for any $r$. To this end, we take a completely different approach -- our key contribution is a novel deterministic variant of the cop decomposition in minor-free graphs [And86, AGG14]. Our shortcut partition for $K_r$-minor-free graphs yields several direct applications. Most notably, we construct the first optimal distance oracle for $K_r$-minor-free graphs, with $1+\varepsilon$ stretch, linear space, and constant query time for any fixed $\varepsilon \in (0,1)$. The previous best distance oracle [AG06] uses $O(n\log n)$ space and $O(\log n)$ query time, and its construction relies on Robertson-Seymour structural theorem and other sophisticated tools. We also obtain the first tree cover of $O(1)$ size for minor-free graphs with stretch $1+\varepsilon$, while the previous best $(1+\varepsilon)$-tree cover has size $O(\log^2 n)$ [BFN19].

翻译：捷径划分是由Chang、Conroy、Le、Milenković、Solomon和Than [CCLMST23]近期提出的一种新型图划分方法，它将图划分为低直径簇。直观而言，捷径划分保证：对于图中任意两个顶点$u$和$v$，存在一条路径仅穿过少数簇。他们证明任意平面图均存在捷径划分，并给出若干应用，包括为任意平面图构造拉伸系数为$1+\varepsilon$且树数目为$O(1)$的树覆盖（对任意固定的$\varepsilon \in (0,1)$）。然而，该构造在多个步骤中严重依赖平面性，因此本质上局限于平面图。本文突破了"平面性障碍"，为任意$r$的$K_r$-无小图子图构造了捷径划分。为此，我们采用了完全不同的方法——核心贡献是提出了无小图子图中Cop分解的新型确定性变体[And86, AGG14]。针对$K_r$-无小图子图的捷径划分直接产生若干应用。最重要的是，我们首次为$K_r$-无小图子图构造了最优距离预言机：拉伸系数为$1+\varepsilon$，线性空间，且对任意固定的$\varepsilon \in (0,1)$具有常数查询时间。此前最优距离预言机[AG06]使用$O(n\log n)$空间和$O(\log n)$查询时间，其构造依赖Robertson-Seymour结构定理及其他复杂工具。我们还首次为无小图子图构造了规模为$O(1)$且拉伸系数为$1+\varepsilon$的树覆盖，而此前最优的$(1+\varepsilon)$-树覆盖规模为$O(\log^2 n)$[BFN19]。