Abstract
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.
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Notes
For ease of exposition, in this paper we use the term set even though the object being referred to might be a multiset.
Note that this edge ordering is not unique.
It may happen that in some rounds no rooted subtrees are colored.
It may happen that both the rooted subtrees \(\mathbf {R},\mathbf {S}\) are matched to different vertices in \(M_{{\bar{G}}_{{\mathcal {P}}_{i-1}[\{u,v\}]\cup {\mathcal {Q}}_i}}\)
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Rawat, A., Shayman, M. A \(\frac{5}{2}\)-approximation algorithm for coloring rooted subtrees of a degree 3 tree. J Comb Optim 40, 69–97 (2020). https://doi.org/10.1007/s10878-020-00564-6
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DOI: https://doi.org/10.1007/s10878-020-00564-6