双参数化快速双聚类算法
Fast Biclustering by Dual Parameterization
摘要：
我们研究了两个聚类问题，Starforest编辑，添加和删除边的问题，以获得一个不相交的联盟的明星，和推广双聚类编辑。我们表明，除了NP-难，没有一个问题可以在次指数时间内解决，除非指数时间假设失败。Misra，Panolan和Saurabh（MFCS 2013）认为，在解决方案中引入连接组件数量的界限不应该使问题变得更容易：特别是，他们认为Fomin等人（J. Comput.系统科学，2014（7）是一个例外，而不是规则。在这里，p是一个次要参数，限制了解决方案中组件的数量。然而，在解决方案中的星或biclique的数量的边界，我们得到的算法，运行在时间$2^{5 \sqrt{pk}} + O(n+m)$为p-Starforest编辑和$2^{O(p \sqrt{k} \log(pk))} + O(n+m)$为p-Bicluster编辑。对于更一般的t-Partite p-Cluster Editing情况，我们得到了类似的结果。对于固定数量的簇，这是k的次指数，因为p被认为是常数。我们的研究结果甚至出了多变量次指数时间算法的数量，并有理由相信这一领域值得进一步研究。
英文摘要：
We study two clustering problems, Starforest Editing, the problem of adding and deleting edges to obtain a disjoint union of stars, and the generalization Bicluster Editing. We show that, in addition to being NP-hard, none of the problems can be solved in subexponential time unless the exponential time hypothesis fails. Misra, Panolan, and Saurabh (MFCS 2013) argue that introducing a bound on the number of connected components in the solution should not make the problem easier: In particular, they argue that the subexponential time algorithm for editing to a fixed number of clusters (p-Cluster Editing) by Fomin et al. (J. Comput. Syst. Sci., 80(7) 2014) is an exception rather than the rule. Here, p is a secondary parameter, bounding the number of components in the solution. However, upon bounding the number of stars or bicliques in the solution, we obtain algorithms which run in time $2^{5 \sqrt{pk}} + O(n+m)$ for p-Starforest Editing and $2^{O(p \sqrt{k} \log(pk))} + O(n+m)$ for p-Bicluster Editing. We obtain a similar result for the more general case of t-Partite p-Cluster Editing. This is subexponential in k for fixed number of clusters, since p is then considered a constant. Our results even out the number of multivariate subexponential time algorithms and give reasons to believe that this area warrants further study.
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