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A \(\frac{5}{2}\)-approximation algorithm for coloring rooted subtrees of a degree 3 tree

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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

  1. For ease of exposition, in this paper we use the term set even though the object being referred to might be a multiset.

  2. Note that this edge ordering is not unique.

  3. It may happen that in some rounds no rooted subtrees are colored.

  4. 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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Authors and Affiliations

  1. Intel Corp., Hillsboro, USA

    Anuj Rawat

  2. University of Maryland, College Park, USA

    Mark Shayman

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  1. Anuj Rawat
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  2. Mark Shayman
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Correspondence to Anuj Rawat.

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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

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