TY - CONF T1 - Algorithmic graph minor theory: Decomposition, approximation, and coloring T2 - Foundations of Computer Science, 2005. FOCS 2005. 46th Annual IEEE Symposium on Y1 - 2005 A1 - Demaine,E. D A1 - Hajiaghayi, Mohammad T. A1 - Kawarabayashi,K. KW - algorithmic graph minor theory KW - approximation algorithm KW - combinatorial polylogarithmic approximation KW - computational complexity KW - constant-factor approximation KW - graph algorithm KW - graph coloring KW - graph colouring KW - half-integral multicommodity flow KW - largest grid minor KW - maximization problem KW - minimization problem KW - polynomial-time algorithm KW - subexponential fixed-parameter algorithm KW - topological graph theory KW - treewidth AB - 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. JA - Foundations of Computer Science, 2005. FOCS 2005. 46th Annual IEEE Symposium on M3 - 10.1109/SFCS.2005.14 ER -