We prove a robust contraction decomposition theorem for $H$-minor-free graphs, which states that given an $H$-minor-free graph $G$ and an integer $p$, one can partition in polynomial time the vertices of $G$ into $p$ sets $Z_1,\dots,Z_p$ such that $\operatorname{tw}(G/(Z_i \setminus Z')) = O(p + |Z'|)$ for all $i \in [p]$ and $Z' \subseteq Z_i$. Here, $\operatorname{tw}(\cdot)$ denotes the treewidth of a graph and $G/(Z_i \setminus Z')$ denotes the graph obtained from $G$ by contracting all edges with both endpoints in $Z_i \setminus Z'$. Our result generalizes earlier results by Klein [SICOMP 2008] and Demaine et al. [STOC 2011] based on partitioning $E(G)$, and some recent theorems for planar graphs by Marx et al. [SODA 2022], for bounded-genus graphs (more generally, almost-embeddable graphs) by Bandyapadhyay et al. [SODA 2022], and for unit-disk graphs by Bandyapadhyay et al. [SoCG 2022]. The robust contraction decomposition theorem directly results in parameterized algorithms with running time $2^{\widetilde{O}(\sqrt{k})} \cdot n^{O(1)}$ or $n^{O(\sqrt{k})}$ for every vertex/edge deletion problems on $H$-minor-free graphs that can be formulated as Permutation CSP Deletion or 2-Conn Permutation CSP Deletion. Consequently, we obtain the first subexponential-time parameterized algorithms for Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Subset Group Feedback Vertex Set, 2-Conn Component Order Connectivity on $H$-minor-free graphs. For other problems which already have subexponential-time parameterized algorithms on $H$-minor-free graphs (e.g., Odd Cycle Transversal, Vertex Multiway Cut, Vertex Multicut, etc.), our theorem gives much simpler algorithms of the same running time.

翻译：我们证明了针对无H-子式图的鲁棒收缩分解定理：给定无H-子式图G和整数p，可在多项式时间内将G的顶点划分为p个集合Z₁,...,Zₚ，使得对于所有i∈[p]和Z'⊆Zᵢ，满足tw(G/(Zᵢ\Z')) = O(p + |Z'|)。其中tw(·)表示图的树宽，G/(Zᵢ\Z')表示通过收缩Zᵢ\Z'中所有边的端点得到的图。我们的结果推广了Klein [SICOMP 2008]和Demaine等人[STOC 2011]基于E(G)划分的早期结论，并涵盖了Marx等人[SODA 2022]针对平面图、Bandyapadhyay等人[SODA 2022]针对有界亏格图（更一般地，几乎可嵌入图）以及Bandyapadhyay等人[SoCG 2022]针对单位圆盘图的最新定理。该鲁棒收缩分解定理可直接推导出在无H-子式图上针对可表述为Permutation CSP Deletion或2-Conn Permutation CSP Deletion的顶点/边删除问题，其参数化算法的时间复杂度为2^{\widetilde{O}(√k)}·n^{O(1)}或n^{O(√k)}。由此，我们首次在无H-子式图上获得了子集反馈顶点集、子集奇环横截集、子集群反馈顶点集、2-连通分量阶连通度等问题的亚指数时间参数化算法。对于已在无H-子式图上存在亚指数时间参数化算法的问题（如奇环横截集、顶点多路割、顶点多割等），本定理提供了具有相同时间复杂度但更为简洁的算法。