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96
An Approximation Algorithm for MinimumCost VertexConnectivity Problems
, 1997
"... We present an approximation algorithm for solving graph problems in which a lowcost set of edges must be selected that has certain vertexconnectivity properties. In the survivable network design problem, one is given a value r ij for each pair of vertices i and j, and must find a minimumcost set ..."
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Cited by 54 (5 self)
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We present an approximation algorithm for solving graph problems in which a lowcost set of edges must be selected that has certain vertexconnectivity properties. In the survivable network design problem, one is given a value r ij for each pair of vertices i and j, and must find a minimumcost set of edges such that there are r ij vertexdisjoint paths between vertices i and j. In the case for which r ij 2 f0; 1; 2g for all i; j, we can find a solution of cost no more than 3 times the optimal cost in polynomial time. In the case in which r ij = k for all i; j, we can find a solution of cost no more than 2H(k) times optimal, where H(n) = 1 + 1 2 + \Delta \Delta \Delta + 1 n . No approximation algorithms were previously known for these problems. Our algorithms rely on a primaldual approach which has recently led to approximation algorithms for many edgeconnectivity problems. 1 Introduction Let G = (V; E) be an undirected graph with nonnegative costs c e 0 on all edges e 2 E. In...
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.
Computing NearOptimal Solutions to Combinatorial Optimization Problems
 IN COMBINATORIAL OPTIMIZATION, DIMACS SERIES IN DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE
, 1995
"... In the past few years, there has been significant progress in our understanding of the extent to which nearoptimal solutions can be efficiently computed for NPhard combinatorial optimization problems. This paper surveys these recent developments, while concentrating on the advances made in the ..."
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Cited by 32 (0 self)
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In the past few years, there has been significant progress in our understanding of the extent to which nearoptimal solutions can be efficiently computed for NPhard combinatorial optimization problems. This paper surveys these recent developments, while concentrating on the advances made in the design and analysis of approximation algorithms, and in particular, on those results that rely on linear programming and its generalizations.
On the equivalence between the primaldual schema and the local ratio technique
 In 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX). Number 2129 in Lecture Notes in Computer Science
, 2001
"... Abstract. We discuss two approximation paradigms that were used to construct many approximation algorithms during the last two decades, the primaldual schema and the local ratio technique. Recently, primaldual algorithms were devised by first constructing a local ratio algorithm and then transform ..."
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Cited by 31 (8 self)
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Abstract. We discuss two approximation paradigms that were used to construct many approximation algorithms during the last two decades, the primaldual schema and the local ratio technique. Recently, primaldual algorithms were devised by first constructing a local ratio algorithm and then transforming it into a primaldual algorithm. This was done in the case of the 2approximation algorithms for the feedback vertex set problem and in the case of the first primaldual algorithms for maximization problems. Subsequently, the nature of the connection between the two paradigms was posed as an open question by Williamson [Math. Program., 91 (2002), pp. 447–478]. In this paper we answer this question by showing that the two paradigms are equivalent.
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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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.
Approximating Steiner Networks with Node Weights
, 2007
"... The (undirected) Steiner Network problem is: given a graph G = (V, E) with edge/node weights and connectivity requirements {r(u, v) : u, v ∈ U ⊆ V}, find a minimum weight subgraph H of G containing U so that the uvedgeconnectivity in H is at least r(u, v) for all u, v ∈ U. The seminal paper of Jai ..."
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The (undirected) Steiner Network problem is: given a graph G = (V, E) with edge/node weights and connectivity requirements {r(u, v) : u, v ∈ U ⊆ V}, find a minimum weight subgraph H of G containing U so that the uvedgeconnectivity in H is at least r(u, v) for all u, v ∈ U. The seminal paper of Jain [19], and numerous papers preceding it, considered the EdgeWeighted Steiner Network problem, with weights on the edges only, and developed novel tools for approximating minimum weight edgecovers of several types of set functions and families. However, for the NodeWeighted Steiner Network (NWSN) problem, nontrivial approximation algorithms were known only for 0, 1 requirements. We make an attempt to change this situation, by giving the first nontrivial approximation algorithm for NWSN with arbitrary requirements. Our approximation ratio for NWSN is rmax · O(ln U), where rmax = maxu,v∈U r(u, v). This generalizes the result of Klein and Ravi [21] for the case rmax = 1. We also give an O(ln U)approximation algorithm for the nodeconnectivity variant of NWSN (when the paths are required to be internallydisjoint) for the case rmax = 2. Our results are based on a much more general approximation algorithm for the problem of finding a minimum nodeweighted edgecover of an uncrossable setfamily. We also give the first evidence that a polylogarithmic approximation ratio for NWSN might not exist even for U  = 2 and unit weights. 1 1
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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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.
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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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.
An approximation algorithm for the fault tolerant metric facility location problem
 Proceedings of APPROX, LNCS
, 1913
"... Abstract. We consider a fault tolerant version of the metric facility location problem in which every city, j, is required to be connected to rj facilities. We give the first nontrivial approximation algorithm for this problem, having an approximation guarantee of 3 · Hk, where k is the maximum re ..."
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Abstract. We consider a fault tolerant version of the metric facility location problem in which every city, j, is required to be connected to rj facilities. We give the first nontrivial approximation algorithm for this problem, having an approximation guarantee of 3 · Hk, where k is the maximum requirement and Hk is the kth harmonic number. Our algorithm is along the lines of [2] for the generalized Steiner network problem. It runs in phases, and each phase, using a generalization of the primaldual algorithm of
An almost O(log k)approximation for kconnected subgraphs
, 2009
"... We consider two cases of the Survivable Network Design (SND) problem: given a complete graph Gn = (V, En) with costs on the edges and connectivity requirements {r(u, v) : u, v ∈ V}, find a minimum cost subgraph G of Gn that contains r(u, v) internally disjoint uvpaths for all u, v ∈ V. Our main res ..."
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We consider two cases of the Survivable Network Design (SND) problem: given a complete graph Gn = (V, En) with costs on the edges and connectivity requirements {r(u, v) : u, v ∈ V}, find a minimum cost subgraph G of Gn that contains r(u, v) internally disjoint uvpaths for all u, v ∈ V. Our main result is an O log k · log n n−k approximation algorithm for the kConnected Subgraph problem (the case r(u, v) = k for all u, v ∈ V), for both directed and undirected graphs, where n = V . Our ratio is O(log k), unless k = n−o(n). Previously, the best known approximation guarantees for this problem were O(log 2 k) for directed/undirected graphs [Kortsarz and Nutov STOC 2004, Fakcharoenphol and Laekhanukit STOC 2008], and O(log k) for undirected graphs with k ≤ √ n/2 [Cheriyan, Vempala, and Vetta STOC 2002]. As in previous work, we consider the kConnectivity Augmentation problem of increasing at minimum cost the connectivity of a given graph J from k − 1 to k; a ρapproximation for it is used to derive an O(ρ · log k)approximation for kConnected Subgraph. Fakcharoenphol and Laekhanukit showed that kConnectivity Augmentation admits an O(log ν)approximation algorithm, where ν is the number of minimal ”violated ” sets in J. However, we may have ν = Θ(n), so this gives only an O(log n)approximation. We design a novel primaldual algorithm that adds an edge set of cost ≤ opt to get ν ≤ 2n n−k. Combined with the algorithm of Fakcharoenphol and Laekhanukit, this gives the ratio O log