# Algorithmic graph minor theory: Decomposition, approximation, and coloring

Title | Algorithmic graph minor theory: Decomposition, approximation, and coloring |

Publication Type | Conference Papers |

Year of Publication | 2005 |

Authors | Demaine ED, Hajiaghayi MT, Kawarabayashi K |

Conference Name | Foundations of Computer Science, 2005. FOCS 2005. 46th Annual IEEE Symposium on |

Date Published | 2005/// |

Keywords | algorithmic graph minor theory, approximation algorithm, combinatorial polylogarithmic approximation, computational complexity, constant-factor approximation, graph algorithm, graph coloring, graph colouring, half-integral multicommodity flow, largest grid minor, maximization problem, minimization problem, polynomial-time algorithm, subexponential fixed-parameter algorithm, topological graph theory, treewidth |

Abstract | 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. |

DOI | 10.1109/SFCS.2005.14 |