Local Tree-Width, Excluded Minors, and Approximation Algorithms

The local tree-width of a graph G = (V;E) is the function ltwG : N ! N that associates with every r 2 N the maximal tree-width of an r-neighborhood in G. Our main graph theoretic result is a decomposition theorem for graphs with excluded minors, which says that such graphs can be decomposed into trees of graphs of almost bounded local tree-width. As an application of this theorem, we show that a number of combinatorial optimization problems, such as MINIMUM VERTEX COVER, MINIMUM DOMINATING SET, and MAXIMUM INDEPENDENT SET have a polynomial time approximation scheme when restricted to a class of graphs with an excluded minor.