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

2025-07-14

AI Summary

This paper studies the Max-Cut problem on interval and split graphs, presenting the first polynomial-time algorithm that strictly improves upon the Goemans–Williamson (GW) approximation ratio α_GW ≈ 0.878, achieving α_GW + ε for ε ≥ 10⁻³⁴. Methodologically, it builds upon the SDP relaxation and randomized rounding framework, deeply integrating structural properties of these graph classes—particularly their interval representations and clique structure—and establishes a novel connection between triangle packings and Max-Cut via edge-triangle covering relations to refine the rounding scheme. Key contributions include: (1) the first rigorous improvement over the GW bound on a nontrivial graph class; (2) a tight quantitative improvement in the approximation ratio; and (3) under the Small Set Expansion Hypothesis, a hardness result showing no (1−c)-approximation algorithm exists for Max-Cut on split graphs, thereby establishing a computational complexity lower bound.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

We present a polynomial-time $(α_{GW} + varepsilon)$-approximation algorithm for the Maximum Cut problem on interval graphs and split graphs, where $α_{GW} approx 0.878$ 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.

Problem

• Improving Maximum Cut approximation for interval graphs
• Enhancing Goemans-Williamson algorithm for split graphs
• Proving hardness of Maximum Cut approximation on split graphs

Innovation

• Improved Goemans-Williamson algorithm analysis
• Triangle packing in interval and split graphs
• Structural results for maximum cut approximation