Thin graph classes and polynomial-time approximation schemes

TL;DR

This paper introduces a new decomposition method called thin systems of overlays, extending Baker's technique to a broader range of graph classes, enabling simpler polynomial-time approximation schemes for complex problems.

Contribution

It presents a novel graph decomposition approach that generalizes Baker's technique, applicable to many minor-closed and subgraph-closed graph classes.

Findings

Many graph classes admit thin system of overlays decompositions.

The new approach simplifies the design of approximation schemes.

Applicable to proper minor-closed and subgraph-closed classes with bounded degree.

Abstract

Baker devised a powerful technique to obtain approximation schemes for various problems restricted to planar graphs. Her technique can be directly extended to various other graph classes, among the most general ones the graphs avoiding a fixed apex graph as a minor. Further generalizations (e.g., to all proper minor closed graph classes) are known, but they use a combination of techniques and usually focus on somewhat restricted classes of problems. We present a new type of graph decompositions (thin systems of overlays) generalizing Baker's technique and leading to straightforward polynomial-time approximation schemes. We also show that many graph classes (all proper minor-closed classes, and all subgraph-closed classes with bounded maximum degree and strongly sublinear separators) admit such decompositions.