彩色分量问题的参数化复杂性与近似问题
Parameterized Complexity and Approximation Issues for the Colorful Components Problems
摘要：
在过去的十年里，对顶点着色图中的彩色分量（每种颜色最多与一个顶点相关联的连通分量）的研究得到了广泛的考虑。在这里，我们考虑两个变体，最小彩色组件（MCC）和最大边传递闭包（MEC），在2011年在生物信息学中的直向基因识别的背景下引入。MCC和MEC的输入都是顶点着色图。MCC要求移除边的子集，使得得到的图被划分为最小数量的彩色连通分量; MEC要求移除边的子集，使得得到的图被划分为彩色连通分量，并且这样的图的传递闭包中的边的数量被最大化。我们研究了MCC和MEC的参数化和近似复杂性，一般和限制的情况下。对于树上的MCC，我们证明了这个问题基本上等价于树上的最小割，因此MCC在因子$1.36 - \varepsilon$内是不可逼近的，它是固定参数可处理的，并且它允许一个多核（当参数是彩色分量的数量时）。此外，我们表明，MCC，虽然它是多项式时间可解的路径，它是NP-困难的，即使是图与不相交的路径数的距离恒定。然后，我们考虑MEC的参数化复杂性时，参数化的解决方案（通过删除边缘，使其在彩色连接组件的分区获得的图）的传递闭包的边缘数$k$。本文给出了一个以$k$为参数的MEC的固定参数算法，当输入图是树时，给出了一个多核。
英文摘要：
The quest for colorful components (connected components where each color is associated with at most one vertex) inside a vertex-colored graph has been widely considered in the last ten years. Here we consider two variants, Minimum Colorful Components (MCC) and Maximum Edges in transitive Closure (MEC), introduced in 2011 in the context of orthology gene identification in bioinformatics. The input of both MCC and MEC is a vertex-colored graph. MCC asks for the removal of a subset of edges, so that the resulting graph is partitioned in the minimum number of colorful connected components; MEC asks for the removal of a subset of edges, so that the resulting graph is partitioned in colorful connected components and the number of edges in the transitive closure of such a graph is maximized. We study the parameterized and approximation complexity of MCC and MEC, for general and restricted instances. For MCC on trees we show that the problem is basically equivalent to Minimum Cut on Trees, thus MCC is not approximable within factor $1.36 - \varepsilon$, it is fixed-parameter tractable and it admits a poly-kernel (when the parameter is the number of colorful components). Moreover, we show that MCC, while it is polynomial time solvable on paths, it is NP-hard even for graphs with constant distance to disjoint paths number. Then we consider the parameterized complexity of MEC when parameterized by the number $k$ of edges in the transitive closure of a solution (the graph obtained by removing edges so that it is partitioned in colorful connected components). We give a fixed-parameter algorithm for MEC paramterized by $k$ and, when the input graph is a tree, we give a poly-kernel.
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