Algorithmic graph minor theory: Decomposition, approximation, and coloring

At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomial time algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a clique-sum of pieces almost-embeddable into bounded-genus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2-approximation to graph coloring, constant-factor approximations to treewidth and the largest grid minor, combinatorial polylogarithmic-approximation to half-integral multicommodity flow, subexponential fixed-parameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor.