Polynomial bounds for the Grid-Minor Theorem

One of the key results in Robertson and Seymour's seminal work on graph minors is the Grid-Minor Theorem (also called the Excluded Grid Theorem). The theorem states that for every fixed-size grid H, every graph whose treewidth is large enough, contains H as a minor. This theorem has found many applications in graph theory and algorithms. Let f(κ) denote the largest value, such that every graph of treewidth κ contains a grid minor of size f(κ) × f(κ). The best current quantitative bound, due to recent work of Kawarabayashi and Kobayashi [15], and Leaf and Seymour [18], shows that f(κ) = Ω(√ log κ/ log log κ). In contrast, the best known upper bound implies that f(κ) = O(√ κ/ log κ) [22]. In this paper we obtain the first polynomial relationship between treewidth and grid-minor size by showing that f(κ) = Ω(k ^δ) for some fixed constant δ > 0, and describe an algorithm, whose running time is polynomial in |V (G)| and κ, that finds a model of such a grid-minor in G.