Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring

  • Authors:

  • Erik D. Demaine;Mohammad Taghi Hajiaghayi;Ken-ichi Kawarabayashi

  • Affiliations:

  • Erik D. Demaine;Mohammad Taghi Hajiaghayi;Ken-ichi Kawarabayashi

  • Venue:
  • FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
  • Year:
  • 2005

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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 polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a clique-sum of pieces almost-embeddable into boundedgenus 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 polylogarithmicapproximation to half-integral multicommodity flow, subexponential fixed-parameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor.