Abstract

We study a version of the graph \(2\)-clustering problem and the related semi-supervised problem. In these problems, given an undirected graph, we have to find a nearest \(2 \)-cluster graph, i.e. a graph on the same vertex set with exactly two nonempty connected components each of which is a complete graph. The distance between two graphs is the number of noncoinciding edges. The problems under consideration are NP-hard. In 2008, Coleman, Saunderson, and Wirth presented a polynomial time \(2 \)-approximation algorithm for the analogous problem in which the number of clusters does not exceed \(2\). Unfortunately, the method of proving the performance guarantee of the Coleman, Saunderson, and Wirth algorithm is inappropriate for the graph \(2\)-clustering problem in which the number of clusters equals \(2\). We propose a polynomial time \(2 \)-approximation algorithm for the \(2 \)-clustering problem on an arbitrary graph. In contrast to the proof by Coleman, Saunderson, and Wirth, our proof of the performance guarantee of this algorithm does not use switchings. Moreover, we propose an analogous \(2 \)-approximation algorithm for the related semi-supervised problem.

中文翻译：

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两个图聚类问题的2-逼近算法

摘要

我们研究了图\（2 \）-聚类问题和相关的半监督问题的版本。在这些问题中，给定无向图，我们必须找到最接近的\（2 \）-簇图，即在同一顶点集上的图，其中恰好有两个非空连接的分量，每个分量都是一个完整的图。两个图之间的距离是不重合边的数量。正在考虑的问题是NP难题。在2008年，Coleman，Saunderson和Wirth针对簇数不超过\（2 \）的类似问题提出了多项式时间\（2 \） -近似算法。不幸的是，证明Coleman，Saunderson和Wirth算法的性能保证的方法不适用于图的\（2 \） -聚类问题，在该图中簇数等于\（2 \）。我们提出了一个多项式时间\（2 \）近似算法为\（2 \）上的任意图形-clustering问题。与Coleman，Saunderson和Wirth的证明相反，我们对这种算法的性能保证的证明不使用开关。此外，针对相关的半监督问题，我们提出了类似的\（2 \） -逼近算法。