Max Cut with Small-Dimensional SDP Solutions

2026-04-15

AI Summary

This work addresses the Max-Cut problem by investigating whether the Goemans–Williamson approximation ratio α_GW ≈ 0.87856 can be surpassed under the assumption that the optimal solution of the standard semidefinite programming (SDP) relaxation lies in a fixed low-dimensional space and satisfies triangle inequalities. To this end, we propose a novel randomized rounding algorithm based on the signs of low-dimensional Gaussian projections and establish a corresponding geometric anti-concentration lemma. Our approach yields, for any fixed dimension d, a polynomial-time algorithm achieving an approximation ratio strictly better than α_GW, with the expected cut value at least (α_GW + 2^{-O(d)}) times the SDP optimum—demonstrating significant performance gains in small dimensions.

Technology Category

Search and Optimization: Non-convex OptimizationConstraint Satisfaction and Optimization: SatisfiabilityReasoning under Uncertainty: Stochastic Optimization

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Large-scale security measurementsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study the Max-Cut semidefinite programming (SDP) relaxation in the regime where a near-optimal solution admits a low-dimensional realization. While the Goemans--Williamson hyperplane rounding achieves the worst-case optimal approximation ratio $α_{GW}\approx 0.87856$, it is natural to ask whether one can beat $α_{GW}$ when the SDP solution lives in $\mathbb{R}^d$ for a small dimension $d$. We answer this in the affirmative for every fixed $d$: there is a polynomial-time rounding algorithm that, given a $d$-dimensional feasible solution to the standard Max-Cut SDP strengthened with triangle inequalities, produces a cut of expected value at least $(α_{GW}+2^{-O(d)})$ times the SDP value. Our improvement is driven by a new geometric anti-concentration lemma for signs of low-dimensional Gaussian projections.

Problem

• Max-Cut
• semidefinite programming
• low-dimensional
• approximation ratio
• Goemans–Williamson

Innovation

• Max-Cut
• semidefinite programming
• low-dimensional rounding
• Goemans–Williamson
• anti-concentration