多色图的实现问题
The Multicolored Graph Realization Problem
摘要：
我们介绍了多色图实现问题（MGRP）。问题的输入是一个彩色图$(G,φ)$，即，一个图及其顶点的着色。我们可以将每个有色图关联到一个集群图（$G_φ)$），其中，在将所有具有相同颜色的顶点折叠到一个节点后，我们删除多个边和自循环。当$S$恰好有一个顶点来自每个颜色类时，顶点集$S$是多色的。问题是要确定是否存在多色集合$S$，使得在用其颜色类识别$S$中的每个顶点之后，$G[S]$与$G_φ$重合。MGR问题是一类广义网络问题，其中大多数是NP-困难的。例如广义MST问题。MGRP是多色团问题的一个推广，当用颜色的数量参数化时，它是W[1]-困难的。因此，MGRP仍然是W[1]-硬的，当参数化的集群图的大小和参数化的任何图形参数$G_φ$，其中的树宽。我们期待的实例中的问题，其中的颜色类和树宽度的$G_φ$是无界的。我们表明，MGRP是NP-完全时，$G_φ$是弦，双凸二分，完全二分或2维网格。我们的硬度结果如下从1-3单调SAT问题的适当减少。我们的减少表明，即使在一个颜色类中的顶点的最大数量是3时，问题仍然很难。在网格的情况下，硬度也保持与有界度的图。我们补充这些结果，显示组合参数化下，MGR问题变得易于处理。
英文摘要：
We introduce the Multicolored Graph Realization problem (MGRP). The input to the problem is a colored graph $(G,φ)$, i.e., a graph together with a coloring on its vertices. We can associate to each colored graph a cluster graph ($G_φ)$ in which, after collapsing to a node all vertices with the same color, we remove multiple edges and self-loops. A set of vertices $S$ is multicolored when $S$ has exactly one vertex from each color class. The problem is to decide whether there is a multicolored set $S$ such that, after identifying each vertex in $S$ with its color class, $G[S]$ coincides with $G_φ$. The MGR problem is related to the class of generalized network problems, most of which are NP-hard. For example the generalized MST problem. MGRP is a generalization of the Multicolored Clique Problem, which is known to be W[1]-hard when parameterized by the number of colors. Thus MGRP remains W[1]-hard, when parameterized by the size of the cluster graph and when parameterized by any graph parameter on $G_φ$, among those for treewidth. We look to instances of the problem in which both the number of color classes and the treewidth of $G_φ$ are unbounded. We show that MGRP is NP-complete when $G_φ$ is either chordal, biconvex bipartite, complete bipartite or a 2-dimensional grid. Our hardness results follows from suitable reductions from the 1-in-3 monotone SAT problem. Our reductions show that the problem remains hard even when the maximum number of vertices in a color class is 3. In the case of the grid, the hardness holds also graphs with bounded degree. We complement those results by showing combined parameterizations under which the MGR problem became tractable.
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