Polynomial-time Approximation of Independent Set Parameterized by Treewidth

TL;DR

This paper demonstrates how any polynomial-time approximation algorithm for maximum independent set with a certain ratio can be transformed into an approximation algorithm based on the graph's treewidth, linking approximation quality to structural graph parameters.

Contribution

It establishes a method to convert general approximation algorithms into treewidth-dependent algorithms, connecting approximation ratios to graph structure.

Findings

Transforms approximation algorithms into treewidth-based algorithms

Provides a new approximation ratio bound depending on treewidth

Combines with existing algorithms to improve approximation guarantees

Abstract

We prove the following result about approximating the maximum independent set in a graph. Informally, we show that any approximation algorithm with a ``non-trivial'' approximation ratio (as a function of the number of vertices of the input graph $G$) can be turned into an approximation algorithm achieving almost the same ratio, albeit as a function of the treewidth of $G$. More formally, we prove that for any function $f$, the existence of a polynomial time $(n/f(n))$-approximation algorithm yields the existence of a polynomial time $O(tw \cdot\log{f(tw)}/f(tw))$-approximation algorithm, where $n$ and $tw$ denote the number of vertices and the width of a given tree decomposition of the input graph. By pipelining our result with the state-of-the-art $O(n \cdot (\log \log n)^2/\log^3 n)$-approximation algorithm by Feige (2004), this implies an $O(tw \cdot (\log \log tw)^3/\log^3 tw)$-approximation algorithm.