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... general case where each pair has a connectivity requirement r ij (the number of edge disjoint paths required 2 for the pair (i; j)) and R is the maximum connectivity requirement. Very recently, Jain =-=[15]-=- gave a factor 2 approximation algorithm for this problem. In this paper we restrict ourselves to the case where r ij = 1 for all pairs. A fairly easy reduction from the Set cover problem shows that i...

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...2Hk-approximation algorithm. Goemans et al. [39] show that their algorithm extends to the case of weakly supermodular functions f , a generalization of proper functions, when the minimally violated sets can be found in polynomial time (as they can for proper f); a function f is weakly supermodular if f(V ) = 0 and for all A, B ⊆ V , either f(A) + f(B) ≤ f(A ∩B) + f(A ∪ B) or f(A) + f(B) ≤ f(A− B) + f(B − A). Mihail, Shallcross, Dean and Mostrel [58] implemented a variation of this algorithm for use in a telephone network design toolkit, and found that it works well in practice. Recently, Jain [49] gave a non-primal-dual 2-approximation algorithm for (ND) for any weakly supermodular function f (assuming a certain polynomial-time separation oracle for f) by showing 14 that any basic solution to the LP relaxation will always contain some e ∈ E for which xe ≥ 1/2. The performance guarantee is obtained by rounding the value for this edge up to 1, then recursing on the remaining subproblem. Although the performance guarantee of Jain’s algorithm is much stronger than that given for the primal-dual algorithm above, the primal-dual algorithm is still of interest. Jain’s algorithm requires solvi...

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...ed to algorithmic work on two other network design models, namely survivable network design and buy-at-bulk network design, but with some crucial differences. In survivable network design (see, e.g., =-=[10, 13, 23, 24]-=-), we are given a graph�with integer-valued flow requirements���between certain pairs of nodes�and�. We must choose a minimum-cost subgraph of�so that for each pair���, there are at least���edge-disjo...

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...rithms for network design, and there have been some notable successes in the design of networks that satisfy various types of \edge connectivity" requirements, e.g., Goemans and Williamson [11], =-=Jain [13], etc-=-. Fewer results are known on the design of networks subject to \vertex connectivity" requirements. We will focus on the classic problem in this class, namely the minimum cost k-vertex connected s...

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...ces to pick at least one edge crossing the corresponding cut. Considering its linear relaxation, 2-approximation algorithms can be obtained either using primal-dual schemes [20] or iterative rounding =-=[27]-=-. However, there is a simple example of a graph (namely, a cycle) showing an integrality gap of 2 already in the spanning tree case, i.e., when�(see, e.g., [37]). A more promising and well-studied LP ...

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... of fundamental importance in combinatorial optimization and there is a vast literature on problems and results. We refer the reader to [32] for classical results on polynomial time algorithms and to =-=[18, 19, 35, 22, 12]-=- for results and pointers on approximation algorithms. Here we briefly discuss the known results and techniques for some specific problems that are closely related to the problems we consider. Buy-at-...

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...nown. However, for undirected graphs, when the paths are required only to be edge disjoint, an approximation algorithm that produces a solution at most twice the value of an optimal was given by Jain =-=[12]-=-. Henceforth, unless stated otherwise, we consider node connectivity only. A -approximation algorithm for a minimization problem is a polynomial time algorithm that produces a solution of value no mo...