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Approximating minimum cost connectivity problems
 58 in Approximation algorithms and Metaheuristics, Editor
, 2007
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Approximation Algorithms for MinimumCost kVertex Connected Subgraphs
 In 34th Annual ACM Symposium on the Theory of Computing
, 2002
"... We present two new algorithms for the problem of nding a minimumcost kvertex connected spanning subgraph. The rst algorithm works on undirected graphs with at least 6k vertices and achieves an approximation of 6 times the kth harmonic number (which is O(log k)), The second algorithm works o ..."
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Cited by 69 (2 self)
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We present two new algorithms for the problem of nding a minimumcost kvertex connected spanning subgraph. The rst algorithm works on undirected graphs with at least 6k vertices and achieves an approximation of 6 times the kth harmonic number (which is O(log k)), The second algorithm works on any graph (directed or undirected) and gives an O( n=)approximation algorithm for any > 0 and k (1 )n. These algorithms improve on the previous best approximation factor (more than k=2). The latter algorithm also extends to other problems in network design with vertex connectivity requirements. Our main tools are setpair relaxations, a theorem of Mader's (in the undirected case) and iterative rounding (general case).
Iterative Rounding 2Approximation Algorithms for MinimumCost Vertex Connectivity Problems
 J. Comput. Syst. Sci
, 2002
"... The survivable network design problem (SNDP) is the following problem: given an undirected graph and values r ij for each pair of vertices i and j, find a minimumcost subgraph such that there are r ij disjoint paths between vertices i and j. In the edge connected version of this problem (ECSNDP) ..."
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Cited by 44 (0 self)
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The survivable network design problem (SNDP) is the following problem: given an undirected graph and values r ij for each pair of vertices i and j, find a minimumcost subgraph such that there are r ij disjoint paths between vertices i and j. In the edge connected version of this problem (ECSNDP) , these paths must be edgedisjoint. In the vertex connected version of the problem (VCSNDP), the paths must be vertex disjoint. The element connectivity problem (ELCSNDP, or ELC) is a problem of intermediate difficulty.
An Iterative Rounding 2Approximation Algorithm for the Element Connectivity Problem
 In 42nd Annual IEEE Symposium on Foundations of Computer Science
, 2001
"... In the edge connected version of this problem (ECSNDP), these paths must be edgedisjoint. In the vertex connected version of the problem (VCSNDP), the paths must be vertex disjoint. Jain et al. [12] propose a version of the problem intermediate in difficulty to these two, called the element conne ..."
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Cited by 26 (2 self)
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In the edge connected version of this problem (ECSNDP), these paths must be edgedisjoint. In the vertex connected version of the problem (VCSNDP), the paths must be vertex disjoint. Jain et al. [12] propose a version of the problem intermediate in difficulty to these two, called the element connectivity problem (ELCSNDP, or ELC). In this problem, the set of vertices is partitioned into terminals and nonterminals. The edges and nonterminals of the graph are called elements. The values are only specified for pairs of terminals must be element disjoint. Thus if are still connected by a path in the network. These variants of SNDP are all known to be NPhard. The best known approximation algorithm for the ECSNDP has performance guarantee of 2 (due to Jain [11]), and iteratively rounds solutions to a linear programming relaxation of the problem. ELC has a primaldual  approximation algorithm, where (Jain et al. [12]). VCSNDP is not known to have a nontrivial approximation algorithm; however, recently Fleischer [7] has shown how to extend the technique of Jain [11] to give a 2approximation algorithm in the case that ! . She also shows that the same techniques will not work for VCSNDP for more general values of . In this paper we show that these techniques can be extended to a 2approximation algorithm for ELC. This gives the first constant approximation algorithm for a general survivable network design problem which allows node failures.
Network Design for Vertex Connectivity
 In Proceedings of ACM Symposium on Theory of Computing (STOC), 2008. 6 C. Chekuri and
, 2008
"... We study the survivable network design problem (SNDP) for vertex connectivity. Given a graph G(V, E) with costs on edges, the goal of SNDP is to find a minimum cost subset of edges that ensures a given set of pairwise vertex connectivity requirements. When all connectivity requirements are between a ..."
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Cited by 25 (4 self)
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We study the survivable network design problem (SNDP) for vertex connectivity. Given a graph G(V, E) with costs on edges, the goal of SNDP is to find a minimum cost subset of edges that ensures a given set of pairwise vertex connectivity requirements. When all connectivity requirements are between a special vertex, called the source, and vertices in a subset T ⊆ V, called terminals, the problem is called the singlesource SNDP. Our main result is a randomized k O(k2) log 4 napproximation algorithm for singlesource SNDP where k denotes the largest connectivity requirement for any sourceterminal pair. In particular, we get a polylogarithmic approximation for any constant k. Prior to our work, no nontrivial approximation guarantees were known for this problem for any k ≥ 3. We also show that SNDP is k Ω(1)hard to approximate and provide an elementary construction that shows that the wellstudied setpair linear programming relaxation for this problem has an ˜ Ω(k 1/3) integrality gap.
Algorithms for SingleSource Vertex Connectivity
"... In the Survivable Network Design Problem (SNDP) the goal is to find a minimum cost subset of edges that satisfies a given set of pairwise connectivity requirements among the vertices. This general network design framework has been studied extensively and is tied to the development of major algorithm ..."
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Cited by 24 (2 self)
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In the Survivable Network Design Problem (SNDP) the goal is to find a minimum cost subset of edges that satisfies a given set of pairwise connectivity requirements among the vertices. This general network design framework has been studied extensively and is tied to the development of major algorithmic techniques. For the edgeconnectivity version of the problem, a 2approximation algorithm is known for arbitrary pairwise connectivity requirements. However, no nontrivial algorithms are known for its vertex connectivity counterpart. In fact, even highly restricted special cases of the vertex connectivity version remain poorly understood. We study the singlesource kvertex connectivity version of SNDP. We are given a graph G(V, E) with a subset T of terminals and a source vertex s, and the goal is to find a minimum cost subset of edges ensuring that every terminal is kvertex connected to s. Our main result is an O(k log n)approximation algorithm for this problem; this improves upon the recent 2 O(k2) log 4 napproximation. Our algorithm is based on an intuitive rerouting scheme. The analysis relies on a structural result that may be of independent interest: we show that any solution can be decomposed into a disjoint collection of multiplelegged spiders, which are then used to reroute flow from terminals to the source via other terminals. We also obtain the first nontrivial approximation algorithm for the vertexcost version of the same problem, achieving an O(k 7 log 2 n)approximation. 1.
Independence Free Graphs and Vertex Connectivity Augmentation
, 2001
"... Given an undirected graph G and a positive integer k, the kvertexconnectivity augmentation problem is to nd a smallest set F of new edges for which G+F is kvertexconnected. Polynomial algorithms for this problem have been found only for k 4 and a major open question in graph connectivity is w ..."
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Cited by 24 (0 self)
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Given an undirected graph G and a positive integer k, the kvertexconnectivity augmentation problem is to nd a smallest set F of new edges for which G+F is kvertexconnected. Polynomial algorithms for this problem have been found only for k 4 and a major open question in graph connectivity is whether this problem is solvable in polynomial time in general. In this
Approximating connectivity augmentation problems
 In SODA. 176–185
, 2005
"... Let G = (V, E) be an undirected graph and let S ⊆ V. The Sconnectivity λS G (u, v) of a node pair (u, v) in G is the maximum number of uvpaths that no two of them have an edge or a node in S − {u, v} in common. The corresponding Connectivity Augmentation (CA) problem is: given a graph G = (V, E), ..."
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Cited by 17 (11 self)
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Let G = (V, E) be an undirected graph and let S ⊆ V. The Sconnectivity λS G (u, v) of a node pair (u, v) in G is the maximum number of uvpaths that no two of them have an edge or a node in S − {u, v} in common. The corresponding Connectivity Augmentation (CA) problem is: given a graph G = (V, E), a node subset S ⊆ V, and a nonnegative integer requirement function r(u, v) on V ×V, add a minimum size set F of new edges to G so that λS G+F (u, v) ≥ r(u, v) for all (u, v) ∈ V ×V. Three extensively studied particular cases are: the EdgeCA (S = ∅), the NodeCA (S = V), and the ElementCA (r(u, v) = 0 whenever u ∈ S or v ∈ S). A polynomial algorithm for EdgeCA was developed by Frank. In this paper we consider the ElementCA and the NodeCA, that are NPhard even for r(u, v) ∈ {0, 2}. The best known ratios for these problems were: 2 for ElementCA and O(rmax · ln n) for NodeCA, where rmax = maxu,v∈V r(u, v) and n = V . Our main result is a 7/4approximation algorithm for the ElementCA, improving the previously best known 2approximation. For ElementCA with r(u, v) ∈ {0, 1, 2} we give a 3/2approximation algorithm. These approximation ratios are based on a new splittingoff theorem, which implies an improved lower bound on the number of edges needed to cover a skewsupermodular set function. For NodeCA we establish the following approximation threshold: NodeCA with r(u, v) ∈ {0, k} cannot be approximated within O(2log1−ε n) for any fixed ε> 0, unless NP ⊆ DTIME(npolylog(n)).