Local tree-width, excluded minors, and approximation algorithms

TL;DR

This paper introduces a decomposition theorem for graphs with excluded minors based on local tree-width, enabling polynomial-time approximation schemes for several combinatorial problems on such graphs.

Contribution

The paper presents a new decomposition theorem for graphs with excluded minors that leverages local tree-width, facilitating efficient approximation algorithms.

Findings

Graphs with excluded minors can be decomposed into trees of graphs with bounded local tree-width.

Polynomial-time approximation schemes are achievable for problems like Vertex Cover and Independent Set on these graphs.

The decomposition theorem has broad implications for algorithm design on complex graph classes.

Abstract

The local tree-width of a graph G=(V,E) is the function ltw^G: N -> N that associates with every natural number r the maximal tree-width of an r-neighborhood in G. Our main graph theoretic result is a decomposition theorem for graphs with excluded minors that essentially says that such graphs can be decomposed into trees of graphs of 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.