发表机构
Northeastern University（东北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对一般k的相关k聚类问题，提出多项式时间常数因子近似算法，在随机顺序在线模型中证明了Ω(log k)（k为poly(n)时为Ω(log n)）的竞争比下界，并给出受Pivot算法启发的多对数竞争比上界。
AI 中文摘要
相关聚类在过去二十年间因广泛的实际应用，已在多种不同计算模型下得到了深入研究。其目标是计算顶点集的一个划分，使得不一致性的总数——即簇间边的数量与簇内非边的数量之和——最小化。本文研究相关k聚类问题，该问题将簇的总数限制为k个。相关k聚类是NP难问题，尽管已有研究针对常数k提出了多项式时间近似方案，但目前尚无针对一般k的已知结果。\n我们的第一项成果是提出了一种适用于一般k的相关k聚类多项式时间常数因子近似算法。本工作的核心聚焦于更具挑战性的在线场景。注意到对抗式到达下的最优竞争比已知为Ω(n)，我们将研究重点放在被广泛研究的随机顺序模型上：在该模型中，顶点以随机顺序到达，且当一个顶点到达时，它与此前已到达邻居之间的边会被揭示。我们证明了一个出人意料的下界：任何在线算法的竞争比都为Ω(log k)，当k=poly(n)时，该下界可扩展为Ω(log n)。最后，本文的核心成果是给出了相关k聚类竞争比的多对数上界，所使用的算法受经典的相关聚类Pivot算法启发。
英文摘要
Correlation clustering has been extensively studied over the last two decades in many different computational models owing to its wide-ranging practical applications. The goal is to compute a partition of the vertex set such that the total number of disagreements, i.e. the sum of edges between clusters, and non-edges within clusters, is minimized. In this paper, we study correlation $k$-clustering, in which the total number of clusters is restricted to $k$. Correlation $k$-clustering is NP-hard, and while previous work has presented a polynomial time approximation scheme for constant $k$, there are no known results for general $k$. Our first result is a polynomial-time constant-factor approximation algorithm for correlation $k$-clustering for general $k$. The main focus of this work is in the more challenging online setting. Noting that the best competitive ratio under adversarial arrivals is known to be $Ω(n)$, we concentrate on the well-studied random-order model, where vertices arrive in random order and on arrival of a vertex, edges to its earlier-arrived neighbors are revealed. We prove a surprising lower bound of $Ω(\log k)$-competitiveness for any online algorithm, which can be extended to $Ω(\log n)$ when $k = \text{poly}(n)$. Finally, the main result of this paper is a polylogarithmic upper bound on the competitive ratio for correlation $k$-clustering, using an algorithm inspired by the classic Pivot algorithm for correlation clustering.
Comments59 pages, 7 figures, 1 table
