arXiv 2210.021212022-10-05cs.DM

稠密随机交图中求最大团的谱算法

A spectral algorithm for finding maximum cliques in dense random intersection graphs

Filippos Christodoulou, Sotiris Nikoletseas, Christoforos Raptopoulos, Paul Spirakis

摘要：

在随机交叉图$G_{n,m,p}$中，每个$n$顶点通过以概率$p$独立地包括每个标签来选择$m$标签集合的随机子集，并且在具有至少一个共同标签的顶点之间绘制边。在其他应用中，这样的图已经被用于对社交网络进行建模，其中个体对应于顶点，并且各种特征（例如，想法、兴趣）对应于标签;共享至少一个共同特征的个体被连接，并且这通过随机相交图中的边来抽象。本文考虑了当输入图为$G_{n,m,p}$时求最大团的问题。目前的算法，这个问题是成功的概率很高，只有相对稀疏的情况下，离开密集的情况下，大多未探索。我们提出了一种谱算法，寻找大集团，处理顶点根据各自的值在第二大特征向量的邻接矩阵的诱导子图的输入图对应的小集团的共同邻居。利用[15]中的单标签团定理，我们能够构建随机实例，而无需在输入图中外部植入一个大团。特别是，我们使用标签选择来确定最大团，然后通过仅将$G_{n, m, p}$的邻接矩阵作为算法的输入来隐藏标签信息。我们的实验评估表明，我们的谱算法明显优于现有的多项式时间算法，无论是在故障概率和近似保证度量，特别是在密集的制度，从而表明随机相交图的谱特性也可以用来构建其他NP难图论问题的有效算法。

英文摘要：

In a random intersection graph $G_{n,m,p}$, each of $n$ vertices selects a random subset of a set of $m$ labels by including each label independently with probability $p$ and edges are drawn between vertices that have at least one label in common. Among other applications, such graphs have been used to model social networks, in which individuals correspond to vertices and various features (e.g. ideas, interests) correspond to labels; individuals sharing at least one common feature are connected and this is abstracted by edges in random intersection graphs. In this paper, we consider the problem of finding maximum cliques when the input graph is $G_{n,m,p}$. Current algorithms for this problem are successful with high probability only for relatively sparse instances, leaving the dense case mostly unexplored. We present a spectral algorithm for finding large cliques that processes vertices according to respective values in the second largest eigenvector of the adjacency matrix of induced subgraphs of the input graph corresponding to common neighbors of small cliques. Leveraging on the Single Label Clique Theorem from [15], we were able to construct random instances, without the need to externally plant a large clique in the input graph. In particular, we used label choices to determine the maximum clique and then concealed label information by just giving the adjacency matrix of $G_{n, m, p}$ as input to the algorithm. Our experimental evaluation showed that our spectral algorithm clearly outperforms existing polynomial time algorithms, both with respect to the failure probability and the approximation guarantee metrics, especially in the dense regime, thus suggesting that spectral properties of random intersection graphs may be also used to construct efficient algorithms for other NP-hard graph theoretical problems as well.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文