Hermite Brings a Laptop：分析Frieze-Jerrum舍入可改进聚类问题的近似算法

中文摘要 AI 辅助

Frieze-Jerrum舍入是对图划分与聚类问题的SDP松弛进行舍入的标准工具，它利用k个独立高斯向量将节点分配到至多k个簇中。其分析依赖碰撞概率P_k(ρ)，即SDP向量内积为ρ的两个节点被分配到同一簇的概率。当k≥4时，目前尚无P_k的易处理闭式表达式，这使得近似保证难以验证，也阻碍了对更优算法的系统性探索。\n我们提出了一种Hermite系数验证框架，用于推导P_k的精确且易处理的界。借助高斯噪声稳定性的Hermite展开，我们将P_k表示为系数非负的幂级数，将这些系数简化为一维高斯积分，并对其中有限个系数进行验证，从而在整个相关范围内得到P_k的严格界。我们的框架为多个聚类问题提供了更强的多项式时间近似算法。\n对于MaxAgree相关聚类，我们得到了0.7818的近似比，这是二十年来首次超越Swamy（2004）提出的0.7666比率的改进。在难度方面，我们证明标准SDP松弛的整性间隙至多为0.802，且超出该值的近似是Unique Games难的。我们还改进了至多K个簇的变体MaxAgree[K]的已知最优比率（例如K=3时从0.77提升至0.8151）。对于Max K-Cut问题，我们通过Hermite展开的一个结构性质，证明了de Klerk等人（2004）提出的一个猜想，该猜想刻画了所有K≥3时Frieze-Jerrum近似比的特征；我们证明该比率是紧的，并且在K≤16时将其确定到10^-6的精度范围内。最后，我们将模块度最大化的加性近似误差从0.42084（Kawase等人，2021）降至0.3790。

英文摘要

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 $P_k(ρ)$ that two nodes whose SDP vectors have inner product $ρ$ 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 $0.3790$.

发表机构

• Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
• Intesa Sanpaolo（意大利联合圣保罗银行）

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