伪度量加权相关聚类：基于谱预聚类的算法
Pseudometric-Weighted Correlation Clustering via Spectral Preclustering
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中文总结 AI 辅助
本文针对伪度量加权相关聚类问题，提出基于谱预聚类的随机多项式时间算法，实现$(2+\varepsilon)$近似，优于先前$10/3$因子，通过簇-LP框架和常数长度随机游走实现。
中文摘要 AI 辅助
我们研究伪度量加权相关聚类问题，其中每对顶点携带一个非负的不一致权重，且这些权重满足三角不等式。对于每个固定的 $\varepsilon>0$，我们给出一个随机多项式时间的 $(2+\varepsilon)$ 近似算法，改进了先前已知的最佳因子 $10/3$。我们的算法将无权相关聚类的簇-LP框架扩展到伪度量权重。加权设置需要同时控制可接受对的总权重和成对边际中的加权误差。我们的谱预聚类在保持近最优解的同时，将可接受总权重限制为 $\operatorname{poly}(1/\varepsilon)\mathrm{OPT}$，其中 $\mathrm{OPT}$ 是最优聚类成本。一个聚合的托勒密型不等式产生度乘积界和见证簇内随机游走的暖启动，使得构造可以使用常数长度的游走。我们使用具有随机停止时间的相关舍入从有界子簇松弛中采样簇。熵界和三角不等式将加权边际误差归因于可接受对，而不是总输入权重。重复采样和逐原子覆盖修正产生一个显式的可行簇-LP解，支持在多项式多个簇上，其值至多为 $(1+\varepsilon)\mathrm{OPT}$。在重新缩放 $\varepsilon$ 后，因子2舍入给出所声称的近似保证。
英文摘要
We study pseudometric-weighted correlation clustering, where every pair of vertices carries a nonnegative disagreement weight and the weights satisfy the triangle inequality. For every fixed $\varepsilon>0$, we give a randomized polynomial-time $(2+\varepsilon)$-approximation, improving the previously best known factor of $10/3$. Our algorithm extends the cluster-LP framework for unweighted correlation clustering to pseudometric weights. The weighted setting requires controlling both the total weight of admissible pairs and the weighted error in pairwise marginals. Our spectral preclustering preserves a near-optimal solution while bounding the total admissible weight by $\operatorname{poly}(1/\varepsilon)\mathrm{OPT}$, where $\mathrm{OPT}$ is the optimal clustering cost. An aggregated Ptolemy-type inequality yields a degree-product bound and a warm start for random walks within witness clusters, allowing the construction to use walks of constant length. We sample clusters from a bounded sub-cluster relaxation using correlated rounding with a randomized stopping time. An entropy bound and the triangle inequality charge the weighted marginal error to the admissible pairs rather than to the total input weight. Repeated sampling and atom-wise coverage corrections produce an explicit feasible cluster-LP solution supported on polynomially many clusters, with value at most $(1+\varepsilon)\mathrm{OPT}$. After rescaling $\varepsilon$, factor-$2$ rounding gives the stated approximation guarantee.
发表机构
- Seoul National University（首尔大学）
- New York University（纽约大学）
- University of Michigan（密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。
