Jiehua Chen, Hendrik Molter, Manuel Sorge, Ondřej Suchý
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In <span>Temporal Cluster Editing</span> we receive a <i>sequence</i> of layers and we want to transform each layer into a cluster graph so that consecutive layers differ only slightly. That is, we want to transform each layer into a cluster graph with at most <i>k</i> edge additions or deletions and to mark a distinct set of <i>d</i> vertices in each layer so that each two consecutive layers are the same after removing the vertices marked in the first of the two layers. We study the combinatorial structure of the two problems via their parameterized complexity with respect to the parameters <i>d</i> and <i>k</i>, among others. 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引用次数: 0
摘要
最近,对多层图和时序图聚类算法的研究迅速发展,受此激励,我们研究了经典聚类编辑问题的扩展。在多层聚类编辑中,我们会收到一组相同顶点集上的图,称为层,目的是将所有层转化为仅有细微差别的聚类图(小群的不相交联盟)。更具体地说,我们希望最多标记 d 个顶点,并使用每层最多 k 条边的增减将每层转化为聚类图,这样,如果我们移除标记的顶点,就能在所有层中得到相同的聚类图。在 "时间聚类编辑 "中,我们会收到一连串的图层,我们希望将每一层转化为聚类图,这样连续的图层之间只有细微的差别。也就是说,我们希望将每一层转化为最多有 k 条边增删的簇图,并在每一层中标记一组不同的 d 个顶点,这样在去除第一层中标记的顶点后,每两个连续的层都是相同的。我们通过参数 d 和 k 等参数的参数化复杂度来研究这两个问题的组合结构。尽管定义相似,这两个问题的表现却大相径庭:特别是,对于大小为 s 的输入,多层集群编辑是固定参数可处理的,其运行时间为 \(k^{O(k + d)} s^{O(1)}\) ,而时态集群编辑即使在 \(d = 3\) 的情况下,相对于 k 也是\(\textsf {W[1]}\)困难的。
Cluster Editing for Multi-Layer and Temporal Graphs
Motivated by the recent rapid growth of research for algorithms to cluster multi-layer and temporal graphs, we study extensions of the classical Cluster Editing problem. In Multi-Layer Cluster Editing we receive a set of graphs on the same vertex set, called layers and aim to transform all layers into cluster graphs (disjoint unions of cliques) that differ only slightly. More specifically, we want to mark at most d vertices and to transform each layer into a cluster graph using at most k edge additions or deletions per layer so that, if we remove the marked vertices, we obtain the same cluster graph in all layers. In Temporal Cluster Editing we receive a sequence of layers and we want to transform each layer into a cluster graph so that consecutive layers differ only slightly. That is, we want to transform each layer into a cluster graph with at most k edge additions or deletions and to mark a distinct set of d vertices in each layer so that each two consecutive layers are the same after removing the vertices marked in the first of the two layers. We study the combinatorial structure of the two problems via their parameterized complexity with respect to the parameters d and k, among others. Despite the similar definition, the two problems behave quite differently: In particular, Multi-Layer Cluster Editing is fixed-parameter tractable with running time \(k^{O(k + d)} s^{O(1)}\) for inputs of size s, whereas Temporal Cluster Editing is \(\textsf {W[1]}\)-hard with respect to k even if \(d = 3\).
期刊介绍:
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