Polynomial Bounds for the Grid-Minor Theorem

Abstract

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 grid H , every graph whose treewidth is large enough relative to | V ( H )| 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 previous quantitative bound, due to recent work of Kawarabayashi and Kobayashi, and Leaf and Seymour, shows that f ( k )=Ω(√log k /log log k ). In contrast, the best known upper bound implies that f ( k ) = O (√ k /log k ). In this article, 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 a randomized algorithm, whose running time is polynomial in | V ( G )| and k , that with high probability finds a model of such a grid minor in G .

A Polynomial Excluded-Minor Approximation of Treedepth

“…To obtain the algorithmic version of Theorem 1.2, we combine Corollary 6.2 with the randomized polynomial-time algorithm of Chekuri and Chuzhoy [12] which, given a graph G, a minor embedding of the k × k grid where k = tw(G) Ω(1) .…”

“…The question arises whether one (or a bounded number of) uniform families of non-minimal obstructions suffice for a polynomial approximation of a given minormonotone graph invariant. A recent breakthrough of Chekuri and Chuzhoy [12] gave precisely such a result for treewidth (resolving a longstanding conjecture in graph minor theory). Theorem 1.1.…”

“…Due to the constructive nature of the proofs, we get an algorithmic version of Theorem 1.3. Combined with the algorithmic version of Theorem 1.1 from [12], we get a randomized polynomial-time algorithm which, given a graph G, either determines that G is of small treedepth or outputs a certificate of one of the three cases in Theorem 1.2 (i.e., a minor-embedding of a grid, tree, or path. For details see Section 6).…”

“…Understanding these invariants in terms of excluded minors has been an active area of research. The recent Grid Minor Theorem of Chekuri and Chuzhoy [12] establishes that treewidth is polynomially approximated by the largest k × k grid minor. In this paper, we give a similar polynomial excluded-minor approximation for treedepth in terms of three basic obstructions: grids, tree, and paths.…”

“…Theorem 1.1. (Polynomial Grid-Minor Theorem for Treewidth [12]) There is an absolute constant c such that every graph with treewidth ≥ k c has a k × k grid minor.…”