Robust Contraction Decomposition for Minor-Free Graphs and its Applications
Sayan Bandyapadhyay, William Lochet, Daniel Lokshtanov, Dániel Marx, Pranabendu Misra, Daniel Neuen, Saket Saurabh, Prafullkumar Tale, Jie Xue
TL;DR
This work delivers a robust vertex contraction decomposition theorem for $H$-minor-free graphs, enabling partitions into $p$ parts so that contracted torsos maintain treewidth $O(p+|Z'|)$ for any subset $Z' eq Z_i$. Building on Robertson–Seymour decomposition and almost-embeddable pieces, the authors develop a local-to-global framework that carefully handles fake edges and boundary connectivity, culminating in a polynomial-time construction of the partition. The main result directly yields subexponential-time parameterized algorithms for a broad class of vertex/edge deletion problems expressible as Permutation CSP Deletions, including first subexponential algorithms for Subset Feedback Vertex Set, Subset Odd Cycle Transversal, and Subset Group Feedback Vertex Set on $H$-minor-free graphs. The approach also simplifies and unifies prior techniques across planar, bounded-genus, and unit-disk graph settings, contributing a versatile structural tool for subexponential parameterized computation on minor-free graphs.
Abstract
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.