Large-Treewidth Graph Decompositions and Applications

Treewidth is a graph parameter that plays a fundamental role in several structural and algorithmic results. We study the problem of decomposing a given graph $G$ into node-disjoint subgraphs, where each subgraph has sufficiently large treewidth. We prove two theorems on the tradeoff between the number of the desired subgraphs $h$, and the desired lower bound $r$ on the treewidth of each subgraph. The theorems assert that, given a graph $G$ with treewidth $k$, a decomposition with parameters $h,r$ is feasible whenever $hr^2 \le k/\polylog(k)$, or $h^3r \le k/\polylog(k)$ holds. We then show a framework for using these theorems to bypass the well-known Grid-Minor Theorem of Robertson and Seymour in some applications. In particular, this leads to substantially improved parameters in some Erdos-Posa-type results, and faster algorithms for a class of fixed-parameter tractable problems.

💡 Research Summary

The paper investigates the problem of partitioning a graph of treewidth k into h node‑disjoint subgraphs, each of which has treewidth at least r. Two main theorems give sufficient conditions for such a decomposition. Theorem 1.1 states that if hr² ≤ k / polylog k, then an efficient randomized polynomial‑time algorithm can produce the desired partition. Theorem 1.2 relaxes the condition to h³r ≤ k / polylog k, again with an efficient algorithm. The two results trade off the parameters h and r: the first is stronger when r is small and many subgraphs are needed, while the second is better when the number of subgraphs is modest but each must be large. The authors conjecture a unified bound hr ≤ k / polylog k, which would subsume both theorems.

Technically, the proofs combine three ingredients. First, for constant‑degree expanders (which have treewidth Θ(n)), the authors show how to embed many vertex‑disjoint “expander copies” of size Θ(r) while preserving treewidth, using short‑path routing results. Second, for general graphs they repeatedly cut along small separators, preserving large treewidth on both sides, and apply the process recursively until the required number of pieces is obtained. A key tool here is the notion of a contracted graph, which captures a minimal subgraph that retains most of the original treewidth while having a small boundary that is well‑linked to the interior. Third, they use bramble‑based certificates and recent grid‑like minor constructions to certify lower bounds on treewidth without relying on the classical Grid‑Minor Theorem.

The paper then demonstrates two families of applications where these decomposition theorems replace the Grid‑Minor Theorem and lead to substantially better parameters.

1. Erdős‑Pósa‑type results. Classical Erdős‑Pósa theorems (e.g., for cycles of length ≡ 0 (mod m)) typically use the Grid‑Minor Theorem to argue that a graph of treewidth g(k)=2^{O(k)} must contain k disjoint target subgraphs, yielding covering bounds f(k) that are exponential or at best quadratic in k. By applying Theorem 1.1, the authors show that it suffices that treewidth be only \tilde O(k), which reduces the covering function to f(k)=\tilde O(k). This improves several known bounds, including those for planar‑minor families and for families defined by a fixed forbidden minor.

2. Fixed‑parameter tractable (FPT) algorithms. The bidimensionality framework yields FPT algorithms for many problems on planar or H‑minor‑free graphs with running time 2^{O(k²)}·n^{O(1)} because it relies on the Grid‑Minor Theorem. In general graphs the same approach would give 2^{O(k^{2.5})}·n^{O(1)}. Using the new decomposition theorems, the authors obtain algorithms with running time 2^{O(k·polylog k)}·n^{O(1)} for the same class of problems, i.e., a single‑exponential dependence on the parameter. The improvement stems from the ability to break the graph into many large‑treewidth pieces and then apply local arguments (which may still use grid‑like minors) within each piece.

The paper also discusses related work on brambles, well‑linked sets, and grid‑like minors, emphasizing that the new approach is orthogonal to prior techniques that rely on a single global structure. By focusing on a decomposition into many locally large‑treewidth components, the method is agnostic to how the lower bound on treewidth is certified in each component, allowing the use of any available tool (grid minors, grid‑like minors, or brambles).

Finally, the authors outline future directions: proving the conjectured unified bound, tightening the polylogarithmic factors, extending the framework to other parameterized problems (e.g., path cover, feedback vertex set), and experimental evaluation on large networks.

In summary, the paper introduces a novel decomposition paradigm for high‑treewidth graphs, provides efficient algorithms under two concrete trade‑offs, and demonstrates that this paradigm can replace the Grid‑Minor Theorem in several important algorithmic contexts, yielding substantially improved bounds for Erdős‑Pósa‑type theorems and for the running time of FPT algorithms on general graphs.