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(k) denote the largest value, such that every graph of treewidth k contains a grid minor of size f(k) × f(k). The best current quantitative bound, due to recent work of Kawarabayashi and Kobayashi [15], and Leaf and Seymour [18], shows that f(k) = Ω(√logk/loglogk). In contrast, the best known upper bound implies that f(k) = O(√k/logk) [22]. In this paper we obtain the first polynomial relationship between treewidth and grid-minor size by showing that f(k) = Ω(k^δ) for some fixed constant δ > 0, and describe an algorithm, whose running time is polynomial in |V (G)| and k, that finds a model of such a grid-minor in G.