Approximating the Generalized Capacitated Tree-Routing Problem

  • Authors:

  • Ehab Morsy;Hiroshi Nagamochi

  • Affiliations:

  • Department of Applied Mathematics and Physics Graduate School of Informatics, Kyoto University, Yoshida Honmachi, Sakyo, Kyoto, Japan 606-8501;Department of Applied Mathematics and Physics Graduate School of Informatics, Kyoto University, Yoshida Honmachi, Sakyo, Kyoto, Japan 606-8501

  • Venue:
  • COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
  • Year:
  • 2008

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Abstract

In this paper, we introduce the generalized capacitated tree-routing problem(GCTR), which is described as follows. Given a connected graph G= (V,E) with a sink s茂戮驴 Vand a set M茂戮驴 V茂戮驴 {s} of terminals with a nonnegative demand q(v), v茂戮驴 M, we wish to find a collection ${\mathcal T}=\{T_{1},T_{2},\ldots,T_{\ell}\}$ of trees rooted at sto send all the demands to s, where the total demand collected by each tree Tiis bounded from above by a demand capacity 茂戮驴 0. Let 茂戮驴 0 denote a bulk capacity of an edge, and each edge e茂戮驴 Ehas an installation cost w(e) 茂戮驴 0 per bulk capacity; each edge eis allowed to have capacity k茂戮驴for any integer k, which installation incurs cost kw(e). To establish a tree routing Ti, each edge econtained in Tirequires 茂戮驴+ βq茂戮驴 amount of capacity for the total demand q茂戮驴 that passes through edge ealong Tiand prescribed constants 茂戮驴,β茂戮驴 0, where 茂戮驴means a fixed amount used to separate the inside of the routing Tifrom the outside while term βq茂戮驴 means the net capacity proportional to q茂戮驴. The objective of GCTR is to find a collection ${\mathcal T}$ of trees that minimizes the total installation cost of edges. Then GCTR is a new generalization of the several known multicast problems in networks with edge/demand capacities. In this paper, we prove that GCTR is $(2[ \lambda/(\alpha+\beta \kappa)] /\lfloor \lambda/(\alpha+\beta \kappa)\rfloor +\rho_{\mbox{\tiny{\sc ST}}})$-approximable if 茂戮驴茂戮驴 茂戮驴+ β茂戮驴holds, where $\rho_{\mbox{\tiny{\sc ST}}}$ is any approximation ratio achievable for the Steiner tree problem.