多色生成图
Multi-Colored Spanning Graphs
摘要：
我们研究了Hurtado等人提出的一个问题。（2016）由稀疏集可视化激发。给定平面中的$n$点，每个点用一个或多个原色标记，\emph{colored spanning graph}（CSG）是一个图，使得对于每个原色，该颜色的顶点导出连通子图。\textsc{Min-CSG}问题要求着色生成图中的边长度和最小。我们发现，这个问题是NP-难的$k$原色时，$k\ge 3$，并提供了一个$(2-\frac{1}{3+2\varrho})$近似算法的$k=3$，在多项式时间内运行，其中$\varrho$是施泰纳比。在输入点共线且$k$为常数的特殊情况下，给出了一个$O(n)$时间算法.
英文摘要：
We study a problem proposed by Hurtado et al. (2016) motivated by sparse set visualization. Given $n$ points in the plane, each labeled with one or more primary colors, a \emph{colored spanning graph} (CSG) is a graph such that for each primary color, the vertices of that color induce a connected subgraph. The \textsc{Min-CSG} problem asks for the minimum sum of edge lengths in a colored spanning graph. We show that the problem is NP-hard for $k$ primary colors when $k\ge 3$ and provide a $(2-\frac{1}{3+2\varrho})$-approximation algorithm for $k=3$ that runs in polynomial time, where $\varrho$ is the Steiner ratio. Further, we give a $O(n)$ time algorithm in the special case that the input points are collinear and $k$ is constant.
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。
