A rooted tree \(\mathbf {R}\) is a rooted subtree of a tree T if the tree obtained by replacing the directed edges of \(\mathbf {R}\) by undirected edges is a subtree of T. We study the problem of assigning minimum number of colors to a given set of rooted subtrees \({\mathcal {R}}\) of a given tree T such that if any two rooted subtrees share a directed edge, then they are assigned different colors. The problem is NP hard even in the case when the degree of T is restricted to at most 3 (Erlebach and Jansen, in: Proceedings of the 30th Hawaii international conference on system sciences, p 221, 1997). We present a \(\frac{5}{2}\)-approximation algorithm for this problem. The motivation for studying this problem stems from the problem of assigning wavelengths to multicast traffic requests in all-optical WDM tree networks.

中文翻译：

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$$ \ frac {5} {2} $ 52-近似算法，用于为度数为3的树的有根子树着色

如果通过用无向边替换\（\ mathbf {R} \）的有向边获得的树是T的子树，则根树\（\ mathbf {R} \）是树T的根子树。我们研究了将最小数量的颜色分配给给定树T的给定根子树\（{\ mathcal {R}} \）的问题，使得如果任意两个根子树共享有向边，则它们将被分配不同的颜色颜色。即使将T的程度限制为最大3 ，问题也很难解决（Erlebach和Jansen，于：第30届夏威夷国际系统科学会议论文集，第221页，1997年）。我们提出一个\（\ frac {5} {2} \）-这个问题的近似算法。研究此问题的动机源于在全光WDM树网络中为多播流量请求分配波长的问题。