Abstract
In the Survivable Network Design problem (SNDP), we are given an undirected graph $G(V, E)$ with costs on edges, along with a connectivity requirement $r(u, v)$ for each pair $u, v$ of vertices. The goal is to find a minimum-cost subset $E^*$ of edges, that satisfies the given set of pairwise connectivity requirements. In the edge-connectivity version we need to ensure that there are $r(u, v)$ edge-disjoint paths for every pair $u, v$ of vertices, while in the vertex-connectivity version the paths are required to be vertex-disjoint. The edge-connectivity version of SNDP is known to have a 2-approximation. However, no non-trivial approximation algorithm has been known so far for the vertex version of SNDP, except for special cases of the problem. We present an extremely simple algorithm to achieve an $O(k^3 \log |T|)$-approximation for this problem, where $k$ denotes the maximum connectivity requirement, and $T$ is the set of vertices that participate in one or more pairs with non-zero connectivity requirements. We also give a simple proof of the recently discovered $O(k^2 \log |T|)$-approximation algorithm for the single-source version of vertex-connectivity SNDP. Our results establish a natural connection between vertex-connectivity and a well-understood generalization of edge-connectivity, namely, element-connectivity, in that, any instance of vertex-connectivity can be expressed by a small number of instances of the element-connectivity problem.
Highlights
In the Survivable Network Design Problem (SNDP), we are given an undirected graph G(V, E) with costs on edges, and an integer connectivity requirement r(u, v) for each pair u, v of vertices
Kortsarz et al [17] showed that VC-SNDP is hard to approximate to within a factor of 2log1−ε n for any ε > 0, unless NP ⊆ DTIME npolylog(n), when k is polynomially large in n, that is, k = Ω(nδ ) for some constant δ > 0
There is a polynomial-time randomized O(k3 log |T |)-approximation algorithm for VCSNDP, where k is the largest pairwise connectivity requirement. The proof of this result is based on a randomized reduction that maps a given instance of VC-SNDP to a family of instances of Elem-SNDP
Summary
In the Survivable Network Design Problem (SNDP), we are given an undirected graph G(V, E) with costs on edges, and an integer connectivity requirement r(u, v) for each pair u, v of vertices. The goal is to find a minimum cost subset E∗ of edges such that each pair (u, v) of vertices is connected by r(u, v) paths in the subgraph induced by E∗. It is not hard to show that EC-SNDP can be cast as a special case of VC-SNDP (see the full version of [8]). Even for k = 1, both VC-SNDP and EC-SNDP are known to be APX-hard [2]; the two problems are equivalent for k = 1, and this special case is commonly referred to as the minimum Steiner Forest problem
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