Hermite Brings a Laptop: Analyzing Frieze-Jerrum Rounding Yields Improved Approximations for Clustering Problems

2026-09-28

AI Summary

This study addresses the absence of closed-form collision probabilities in Frieze-Jerrum rounding for k≥4, which has long constrained approximation guarantees for clustering problems. To overcome this analytical bottleneck, we construct a Hermite coefficient certification framework that derives Gaussian noise stability expansions to precisely bound collision probabilities. By integrating semidefinite programming relaxations with numerical quadrature certification, we establish the first computable and rigorous bounds on P_k, yielding new theoretical relationships between upper and lower bounds. Consequently, this work achieves a 0.7818 approximation ratio for MaxAgree correlation clustering—the first improvement in two decades—refines the conjecture for Max K-Cut, and optimizes the modularity maximization error to 0.3790.

Abstract

The Frieze-Jerrum rounding is a standard tool for rounding SDP relaxations of graph partitioning and clustering problems, assigning nodes to at most $k$ clusters using $k$ independent Gaussian vectors. Its analysis hinges on the collision probability $P_k(\rho)$ that two nodes whose SDP vectors have inner product $\rho$ are assigned to the same cluster. No tractable closed form for $P_k$ is known for $k\geq 4$, making it difficult to certify approximation guarantees and hindering the systematic search for better algorithms. We develop a Hermite-coefficient certification framework to derive accurate and tractable bounds on $P_k$. Using the Hermite expansion of Gaussian noise stability, we express $P_k$ as a power series with nonnegative coefficients, reduce these coefficients to one-dimensional Gaussian integrals, and certify finitely many of them, yielding rigorous bounds on $P_k$ over the entire correlation range. Our framework yields strengthened polynomial-time approximations for several clustering problems. For MaxAgree Correlation Clustering, we derive a $0.7818$-approximation, the first improvement in two decades over the $0.7666$ ratio of Swamy (2004). On the hardness side, we show that the integrality ratio of the standard SDP relaxation is at most $0.802$, and that approximation beyond that is Unique Games-hard. We also improve the best known ratios for the variant with at most $K$ clusters, MaxAgree$[K]$ (e.g., from $0.77$ to $0.8151$ for $K=3$). For Max $K$-Cut we resolve, via a structural property of the Hermite expansion, a conjecture of de Klerk et al. (2004) characterizing the Frieze--Jerrum approximation ratio for every $K\ge3$; we show that this ratio is tight, and determine it to within $10^{-6}$ accuracy for $K\le16$. Finally, we reduce the additive approximation error for modularity maximization from $0.42084$ (Kawase et al., 2021) to $0.3790$.

Problem

• Clustering problems
• Frieze-Jerrum rounding
• Collision probability
• Approximation guarantees
• SDP relaxations

Innovation

• Hermite expansion
• Frieze-Jerrum rounding
• SDP relaxation
• approximation algorithms
• correlation clustering