Approximating Maximum Cut on Interval Graphs and Split Graphs beyond Goemans-Williamson

TL;DR

This paper introduces a polynomial-time approximation algorithm for Maximum Cut on interval and split graphs that surpasses the Goemans-Williamson guarantee by leveraging structural properties and triangle packings.

Contribution

It provides an improved approximation algorithm for Maximum Cut on specific graph classes and analyzes their structural properties to justify the approximation bounds.

Findings

Achieves a $( ext{approximation factor} + ext{small constant})$-approximation for Max Cut.

Shows no polynomial-time $(1 - c)$-approximation exists for split graphs under the Small Set Expansion Hypothesis.

Provides structural insights into interval and split graphs related to triangle packings.

Abstract

We present a polynomial-time $(\alpha_{GW} + \varepsilon)$-approximation algorithm for the Maximum Cut problem on interval graphs and split graphs, where $\alpha_{GW} \approx 0.878$ is the approximation guarantee of the Goemans-Williamson algorithm and $\varepsilon > 10^{-34}$ is a fixed constant. To attain this, we give an improved analysis of a slight modification of the Goemans-Williamson algorithm for graphs in which triangles can be packed into a constant fraction of their edges. We then pair this analysis with structural results showing that both interval graphs and split graphs either have such a triangle packing or have maximum cut close to their number of edges. We also show that, subject to the Small Set Expansion Hypothesis, there exists a constant $c > 0$ such that there is no polyomial-time $(1 - c)$-approximation for Maximum Cut on split graphs.