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Polynomial-Time Approximation Schemes for the Euclidean Survivable Network Design Problem

  • Artur Czumaj
  • Andrzej Lingas
  • Hairong Zhao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2380)

Abstract

The survivable network design problem is a classical problem in combinatorial optimization of constructing a minimum-cost subgraph satisfying predetermined connectivity requirements. In this paper we consider its geometric version in which the input is a complete Euclidean graph. We assume that each vertex v has been assigned a connectivity requirement r v . The output subgraph is supposed to have the vertex-(or edge-, respectively) connectivity of at least minr v, r u for any pair of vertices v, u.

We present the first polynomial-time approximation schemes (PTAS) for basic variants of the survivable network design problem in Euclidean graphs. We first show a PTAS for the Steiner tree problem, which is the survivable network design problem with r v ∈ 0, 1 for any vertex v. Then, we extend it to include the most widely applied case where r v ∈ 0, 1, 2 for any vertex v. Our polynomial-time approximation schemes work for both vertex-and edge-connectivity requirements in time \( \mathcal{O} \) (n log n), where the constants depend on the dimension and the accuracy of approximation. Finally, we observe that our techniques yield also a PTAS for the multigraph variant of the problem where the edge-connectivity requirements satisfy r v ∈ 0,1,..., k and k = \( \mathcal{O} \) (1).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Artur Czumaj
    • 1
  • Andrzej Lingas
    • 2
  • Hairong Zhao
    • 1
  1. 1.Department of Computer ScienceNew Jersey Institute of TechnologyNewarkUSA
  2. 2.Department of Computer ScienceLund UniversityLundSweden

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