Plane Decompositions as Tools for Approximation

TL;DR

This paper introduces plane decompositions, a new graph decomposition method allowing cycles and planarity, to improve approximation algorithms for problems like maximum independent set.

Contribution

It extends tree decompositions to planar decompositions and provides a polynomial-time approximation algorithm for maximum independent set.

Findings

Plane decompositions generalize tree decompositions with cycles.

A polynomial-time algorithm approximates maximum independent set using plane decompositions.

Plane decompositions enable efficient approximation algorithms for planar graphs.

Abstract

Tree decompositions were developed by Robertson and Seymour. Since then algorithms have been developed to solve intractable problems efficiently for graphs of bounded treewidth. In this paper we extend tree decompositions to allow cycles to exist in the decomposition graph; we call these new decompositions plane decompositions because we require that the decomposition graph be planar. First, we give some background material about tree decompositions and an overview of algorithms both for decompositions and for approximations of planar graphs. Then, we give our plane decomposition definition and an algorithm that uses this decomposition to approximate the size of the maximum independent set of the underlying graph in polynomial time.