A $\frac{5}{2}$-Approximation Algorithm for Coloring Rooted Subtrees of a Degree $3$ Tree

A rooted tree $\vec{R}$ is a rooted subtree of a tree $T$ if the tree obtained by replacing the directed edges of $\vec{R}$ 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 $3$. 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.