Pattern-Sparse Tree Decompositions in $H$-Minor-Free Graphs

2026-03-31

AI Summary

This work addresses the challenge of controlling the distribution of size-$k$ solution patterns—such as paths or packings—within tree decompositions of $H$-minor-free graphs, where traditional methods fail to effectively bound their presence in individual bags. The paper proposes a randomized polynomial-time algorithm that samples an induced subgraph together with a tree decomposition, which, with constant probability, fully preserves any given size-$k$ solution pattern while ensuring each bag contains only $\widetilde{O}(\sqrt{k})$ relevant vertices. This approach achieves the first simultaneous sparsification of both the solution pattern and its distance-$d$ neighborhood in $H$-minor-free graphs, overcoming prior limitations to connected patterns or planar graphs. By integrating graph minor theory, random sampling, and neighborhood control, the resulting structured decomposition enables randomized algorithms running in $2^{\widetilde{O}(\sqrt{k})}n^{O(1)}$ time for problems including Directed $k$-Path, $H$-Packing, and neighborhood-constrained variants such as Partial Dominating Set.

Abstract

Given an $H$-minor-free graph $G$ and an integer $k$, our main technical contribution is sampling in randomized polynomial time an induced subgraph $G'$ of $G$ and a tree decomposition of $G'$ of width $\widetilde{O}(k)$ such that for every $Z\subseteq V(G)$ of size $k$, with probability at least $\left(2^{\widetilde{O}(\sqrt{k})}|V(G)|^{O(1)}\right)^{-1}$, we have $Z \subseteq V(G')$ and every bag of the tree decomposition contains at most $\widetilde{O}(\sqrt{k})$ vertices of $Z$. Having such a tree decomposition allows us to solve a wide range of problems in (randomized) time $2^{\widetilde{O}(\sqrt{k})}n^{O(1)}$ where the solution is a pattern $Z$ of size $k$, e.g., Directed $k$-Path, $H$-Packing, etc. In particular, our result recovers all the algorithmic applications of the pattern-covering result of Fomin et al. [SIAM J. Computing 2022] (which requires the pattern to be connected) and the planar subgraph-finding algorithms of Nederlof [STOC 2020]. Furthermore, for $K_{h,3}$-free graphs (which include bounded-genus graphs) and for a fixed constant $d$, we signficantly strengthen the result by ensuring that not only $Z$ has intersection $\widetilde{O}(\sqrt{k})$ with each bag, but even the distance-$d$ neighborhood $N^d_{G}[Z]$ as well. This extension makes it possible to handle a wider range of problems where the neighborhood of the pattern also plays a role in the solution, such as partial domination problems and problems involving distance constraints.

Problem

• H-minor-free graphs
• pattern-sparse tree decompositions
• subgraph sampling
• distance-constrained problems
• parameterized algorithms

Innovation

• tree decomposition
• H-minor-free graphs
• pattern sparsity
• randomized sampling
• distance-d neighborhood