Routing and coloring for maximal number of trees

  • Authors:

  • Xujin Chen;Xiaodong Hu;Tianping Shuai

  • Affiliations:

  • Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, China;Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, China;Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, China

  • Venue:
  • COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
  • Year:
  • 2005

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Abstract

Let G be a undirected connected graph. Given g groups each being a subset of V(G) and a number of colors, we consider how to find a subgroup of subsets such that there exists a tree interconnecting all vertices in each subset and all trees can be colored properly with given colors (no two trees sharing a common edge receive the same color); the objective is to maximize the number of subsets in the subgroup. This problem arises from the application of multicast communication in all optical networks. In this paper, we first obtain an explicit lower bound on the approximability of this problem and prove Ω(g1−ε)-inapproximability even when G is a mesh. We then propose a simple greedy algorithm that achieves performance ratio $O({\sqrt{|E(G)|}})$, which matches the theoretical bounds.