固定聚类数的相关聚类
Correlation Clustering with a Fixed Number of Clusters
摘要：
我们继续调查的问题，相关聚类或聚类的定性信息，这是一个聚类公式，最近已经研究。这里的基本设置是，我们给出一个n个节点（对应于要聚类的节点）上的完全图作为输入，其边被标记为+（对于相似的项目对）和-（对于不相似的项目对）。因此，我们只有作为输入的定性信息的相似性和项目之间没有定量的距离测量。聚类的质量是根据其协议数来衡量的，协议数简单地说就是它正确分类的边的数量，即它将端点放置在不同聚类中的-边的数量加上它将端点放置在同一聚类中的+边的数量之和。在本文中，我们研究的问题，找到聚类，最大限度地提高协议的数量，和互补的最小化版本，我们寻求聚类，最大限度地减少分歧的数量。我们专注于当集群的数量被规定为一个小常数k的情况。我们的主要结果是，对于每一个k，有一个多项式时间的近似计划，最大限度地提高协议和最小化分歧。(The对于每个k >= 2，问题都是NP难的。主要的技术工作是最小化版本，因为用于最大化协议的PTAS遵循Max k-CUT的属性测试器的路线。相比之下，当集群的数量没有指定时，最小化分歧的问题被证明是APX困难的，即使最大化版本允许PTAS。
英文摘要：
We continue the investigation of problems concerning correlation clustering or clustering with qualitative information, which is a clustering formulation that has been studied recently. The basic setup here is that we are given as input a complete graph on n nodes (which correspond to nodes to be clustered) whose edges are labeled + (for similar pairs of items) and - (for dissimilar pairs of items). Thus we have only as input qualitative information on similarity and no quantitative distance measure between items. The quality of a clustering is measured in terms of its number of agreements, which is simply the number of edges it correctly classifies, that is the sum of number of - edges whose endpoints it places in different clusters plus the number of + edges both of whose endpoints it places within the same cluster. In this paper, we study the problem of finding clusterings that maximize the number of agreements, and the complementary minimization version where we seek clusterings that minimize the number of disagreements. We focus on the situation when the number of clusters is stipulated to be a small constant k. Our main result is that for every k, there is a polynomial time approximation scheme for both maximizing agreements and minimizing disagreements. (The problems are NP-hard for every k >= 2.) The main technical work is for the minimization version, as the PTAS for maximizing agreements follows along the lines of the property tester for Max k-CUT. In contrast, when the number of clusters is not specified, the problem of minimizing disagreements was shown to be APX-hard, even though the maximization version admits a PTAS.
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