On the Approximability of Max-Cut on 3-Colorable Graphs and Graphs with Large Independent Sets

2026-04-11

AI Summary

This work investigates how structural properties of graphs—such as 3-colorability or the presence of large independent sets—affect the approximability of Max-Cut. By introducing a novel semidefinite programming (SDP) relaxation, employing interval arithmetic analysis, and leveraging a variant of the Majority-Is-Stablest theorem, the authors establish that Max-Cut on 3-colorable graphs remains hard to approximate beyond the Goemans–Williamson ratio (α_GW). Moreover, they identify a critical threshold α* on the size of the maximum independent set: when the graph contains an independent set exceeding this proportion, one can design an efficient approximation algorithm that surpasses α_GW. These results uncover a delicate interplay between graph structure and the fundamental limits of Max-Cut approximation.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Max-Cut is a classical graph-partitioning problem where given a graph $G = (V,E)$, the objective is to find a cut $(S,S^c)$ which maximizes the number of edges crossing the cut. In a seminal work, Goemans and Williamson gave an $α_{GW} \approx 0.87856$-factor approximation algorithm for the problem, which was later shown to be tight by the work of Khot, Kindler, Mossel, and O'Donnell. Since then, there has been a steady progress in understanding the approximability at even finer levels, and a fundamental goal in this context is to understand how the structure of the underlying graph affects the approximability of the Max-Cut problem. In this work, we investigate this question by exploring how the chromatic structure of a graph affects the Max-Cut problem. While it is well-known that Max-Cut can be solved perfectly and near-perfectly in $2$-colorable and almost $2$-colorable graphs in polynomial time, here we explore its approximability under much weaker structural conditions such as when the graph is $3$-colorable or contains a large independent set. Our main contributions in this context are as follows: 1. We show Max-Cut is $α_{GW}$-hard to approximate for $3$-colorable graphs. 2. We identify a natural threshold $α^*$ such that the following holds. Firstly, for graphs which contain an independent set of size up to $α^*$, Max-Cut continues to be $α_{GW}$-factor hard to approximate. Furthermore, for any graph that contains an independent set of size $> α^*$, there exists an efficient $>α_{GW}$-approximation algorithm for Max-Cut. Our hardness results are derived using various analytical tools and novel variants of the Majority-Is-Stablest theorem, which might be of independent interest. Our algorithmic results are based on a novel SDP relaxation, which is then rounded and analyzed using interval arithmetic.

Problem

• Max-Cut
• approximability
• 3-colorable graphs
• independent set
• hardness of approximation

Innovation

• Max-Cut
• approximability
• 3-colorable graphs
• independent set
• SDP relaxation