Catching Rats in $H$-minor-free Graphs

2025-06-28

AI Summary

This paper resolves an open problem posed by Wood: whether the treewidth and branchwidth of $H$-subgraph-free graphs excluding a $k imes k$ grid can be bounded by a function $f(t,k)$ that is linear in $k$ and polynomial in $t = |V(H)|$. The authors develop a novel variant of structural theorems, establishing explicit polynomial dependence of treewidth and branchwidth on both $t$ and the Euler characteristic of $H$. This yields the first improved bound of $mathcal{O}(t^2k + t^{2304})$, confirming the polynomial realizability of $f$. Technically, the approach integrates graph minor structure theory with topological graph theory. As a consequence, they design a $(g+varepsilon)$-approximation algorithm with runtime $2^{mathsf{poly}(t)/varepsilon} cdot mathsf{poly}(n)$, which either outputs a branch decomposition certifying the bound or explicitly constructs a $k imes k$ grid minor as a counterexample.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We show that every $H$-minor-free graph that also excludes a $(k imes k)$-grid as a minor has treewidth/branchwidth bounded from above by a function $f(t,k)$ that is linear in $k$ and polynomial in $t := |V(H)|$. Such a result was proven originally by [Demaine & Hajiaghayi, Combinatorica, 2008], where $f$ was indeed linear in $k$. However the dependency in $t$ in this result was non-explicit (and huge). Later, [Kawarabayashi & Kobayashi, JCTB, 2020] showed that this bound can be estimated to be $f(t,k)in 2^{mathcal{O}(tlog t)} cdot k$. Wood recently asked whether $f$ can be pushed further to be polynomial, while maintaining the linearity on $k$. We answer this in a particularly strong sense, by showing that the treewidth/branchwidth of $G$ is in $mathcal{O}(gk + t^{2304}),$ where $g$ is the Euler genus of $H$. This directly yields $f(t,k)= mathcal{O}(t^2k + t^{2304})$. Our methods build on techniques for branchwidth and on new bounds and insights for the Graph Minor Structure Theorem (GMST) due to [Gorsky, Seweryn & Wiederrecht, 2025, arXiv:2504.02532]. In particular, we prove a variant of the GMST that ensures some helpful properties for the minor relation. We further employ our methods to provide approximation algorithms for the treewidth/branchwidth of $H$-minor-free graphs. In particular, for every $varepsilon > 0$ and every $t$-vertex graph $H$ with Euler genus $g$, we give a $(g + varepsilon)$-approximation algorithm for the branchwidth of $H$-minor-free graphs running in $2^{mathsf{poly}(t) / varepsilon} cdot mathsf{poly}(n)$-time. Our algorithms explicitly return either an appropriate branch-decomposition or a grid-minor certifying a negative answer.

Problem

• Bounding treewidth in H-minor-free graphs
• Improving dependency on H size in bounds
• Approximating branchwidth for H-minor-free graphs

Innovation

• Linear and polynomial treewidth bound function
• Improved Graph Minor Structure Theorem insights
• Efficient branchwidth approximation algorithms