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

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).

Research supported in part by NSF grant CCR-0105701, SBR grant No. 421090, and TFR grant 221-99-344.