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78
Biconnectivity Approximations and Graph Carvings
, 1994
"... A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified) ? Unfortunately, the problem is known to be ..."
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Cited by 97 (5 self)
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A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified) ? Unfortunately, the problem is known to be NP hard. We consider the problem of finding a better approximation to the smallest 2connected subgraph, by an efficient algorithm. For 2edge connectivity our algorithm guarantees a solution that is no more than 3 2 times the optimal. For 2vertex connectivity our algorithm guarantees a solution that is no more than 5 3 times the optimal. The previous best approximation factor is 2 for each of these problems. The new algorithms (and their analyses) depend upon a structure called a carving of a graph, which is of independent interest. We show that approximating the optimal solution to within an additive constant is NP hard as well. We also consider the case where the graph has edge weigh...
A primaldual approximation algorithm for generalized Steiner network problems
 Combinatorica
, 1995
"... M.I.T. We present the first polynomialtime approximation algorithm for finding a minimumcost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, also called the survivable network design problem. ..."
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Cited by 95 (19 self)
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M.I.T. We present the first polynomialtime approximation algorithm for finding a minimumcost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, also called the survivable network design problem. If k is the maximum cut requirement of the problem, our solution comes within a factor of 2k of optimal. Our algorithm is primaldual and shows the importance of this technique in designing approximation algorithms. 1
Improved Approximation Algorithms for Uniform Connectivity Problems
 J. Algorithms
"... The problem of finding minimum weight spanning subgraphs with a given connectivity requirement is considered. The problem is NPhard when the connectivity requirement is greater than one. Polynomial time approximation algorithms for various weighted and unweighted connectivity problems are given. Th ..."
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Cited by 80 (3 self)
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The problem of finding minimum weight spanning subgraphs with a given connectivity requirement is considered. The problem is NPhard when the connectivity requirement is greater than one. Polynomial time approximation algorithms for various weighted and unweighted connectivity problems are given. The following results are presented: 1. For the unweighted kedgeconnectivity problem an approximation algorithm that achieves a performance ratio of 1.85 is described. This is the first polynomialtime algorithm that achieves a constant less than 2, for all k. 2. For the weighted kvertexconnectivity problem, a constant factor approximation algorithm is given assuming that the edgeweights satisfy the triangle inequality. This is the first constant factor approximation algorithm for this problem. 3. For the case of biconnectivity, with no assumptions about the weights of the edges, an algorithm that achieves a factor asymptotically approaching 2 is described. This matches the previous best...
Approximation Algorithms for Finding Highly Connected Subgraphs
, 1996
"... Contents 1 Introduction 2 1.1 Outline of Chapter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 EdgeConnectivity Problems 3 2.1 Weighted EdgeConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.2 Unweighted EdgeConnectivity : : : : : ..."
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Cited by 71 (1 self)
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Contents 1 Introduction 2 1.1 Outline of Chapter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 EdgeConnectivity Problems 3 2.1 Weighted EdgeConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.2 Unweighted EdgeConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2.1 2 EdgeConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2.2 EdgeConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3 VertexConnectivity Problems 11 3.1 Weighted VertexConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 3.2 Unweighted VertexConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 3.2.1 2 VertexConnectivity : : : : : : : : : : : : : : : : :
An Efficient Approximation Algorithm for the Survivable Network Design Problem
 IN PROCEEDINGS OF THE THIRD MPS CONFERENCE ON INTEGER PROGRAMMING AND COMBINATORIAL OPTIMIZATION
, 1993
"... The survivable network design problem is to construct a minimumcost subgraph satisfying certain given edgeconnectivity requirements. The first polynomialtime approximation algorithm was given by Williamson et al. [20]. This paper gives an improved version that is more efficient. Consider a graph ..."
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Cited by 57 (6 self)
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The survivable network design problem is to construct a minimumcost subgraph satisfying certain given edgeconnectivity requirements. The first polynomialtime approximation algorithm was given by Williamson et al. [20]. This paper gives an improved version that is more efficient. Consider a graph
Hardness of Approximation for VertexConnectivity NetworkDesign Problems
, 2002
"... In the survivable network design problem (SNDP), the goal is to find a minimumcost spanning subgraph satisfying certain connectivity requirements. We study the vertexconnectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertexdisjoint paths con ..."
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Cited by 52 (4 self)
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In the survivable network design problem (SNDP), the goal is to find a minimumcost spanning subgraph satisfying certain connectivity requirements. We study the vertexconnectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertexdisjoint paths connecting them.
APPROXIMATING THE MINIMUM EQUIVALENT DIGRAPH
, 1995
"... The minimum equivalent graph (MEG) problem is as follows: given a directed graph, find a smallest subset of the edges that maintains all teachability relations between nodes. This problem is NPhard; this paper gives an approximation algorithm achieving a performance guarantee of about 1.64 in poly ..."
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Cited by 43 (2 self)
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The minimum equivalent graph (MEG) problem is as follows: given a directed graph, find a smallest subset of the edges that maintains all teachability relations between nodes. This problem is NPhard; this paper gives an approximation algorithm achieving a performance guarantee of about 1.64 in polynomial time. The algorithm achieves a performance guarantee of 1.75 in the time required for transitive closure. The heart of the MEG problem is the minimum strongly connected spanning subgraph (SCSS) problemthe MEG problem restricted to strongly connected digraphs. For the minimum SCSS problem, the paper gives apractical, nearly lineartime implementation achieving a performance guarantee of 1.75. The algorithm and its analysis are based on the simple idea of contracting long cycles. The analysis applies directly to2EXCHANCE, a general "local improvement" algorithm, showing that its performance guarantee is 1.75.
When cycles collapse: A general approximation technique for constrained twoconnectivity problems
, 1992
"... We present a general approximation technique for a class of network design problems where we seek a network of minimum cost that satisfies certain communication requirements and is resilient to worstcase singlelink failures. Our algorithm runs in O(n² log n) time on a graph with n nodes and out ..."
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Cited by 36 (12 self)
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We present a general approximation technique for a class of network design problems where we seek a network of minimum cost that satisfies certain communication requirements and is resilient to worstcase singlelink failures. Our algorithm runs in O(n² log n) time on a graph with n nodes and outputs a solution of cost at most thrice the optimum. We extend our technique to obtain approximation algorithms for augmenting a given network so as to satisfy certain communication requirements and achieve resilience to singlelink failures. Our technique allows one to find nearly minimumcost twoconnected networks for a variety of connectivity requirements. For example, our result generalizes earlier results on finding a minimumcost twoconnected subgraph of a given edgeweighted graph in [3, 9] and an earlier result on finding a minimumcost subgraph twoconnecting a specified subset of the nodes in [14]. Using our technique, we can also approximately solve for the first time a ...
Inapproximability of the Tutte polynomial
, 2008
"... The Tutte polynomial of a graph G is a twovariable polynomial T(G; x, y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: take as input a graph G, and output a value which is a good approximation to T(G; x, y). Jaeger, V ..."
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Cited by 29 (8 self)
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The Tutte polynomial of a graph G is a twovariable polynomial T(G; x, y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: take as input a graph G, and output a value which is a good approximation to T(G; x, y). Jaeger, Vertigan and Welsh have completely mapped the complexity of exactly computing the Tutte polynomial. They have shown that this is #Phard, except along the hyperbola (x − 1)(y − 1) = 1 and at four special points. We are interested in determining for which points (x, y) there is a fully polynomial randomised approximation scheme (FPRAS) for T(G; x, y). Under the assumption RP = NP, we prove that there is no FPRAS at (x, y) if (x, y) is is in one of the halfplanes x < −1 or y < −1 (excluding the easytocompute cases mentioned above). Two exceptions to this result are the halfline x < −1, y = 1 (which is still open) and the portion of the hyperbola (x − 1)(y − 1) = 2 corresponding to y < −1 which we show
Restoration Algorithms for Virtual Private Networks in the Hose Model
, 2002
"... A Virtual Private Network (VPN) aims to emulate the services provided by a private network over the shared Internet. The endpoints of a VPN are connected using abstractions such as Virtual Channels (VCs) of ATM or Label Switching Paths (LSPs) of MPLS technologies. Reliability of an endtoend VPN co ..."
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Cited by 27 (1 self)
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A Virtual Private Network (VPN) aims to emulate the services provided by a private network over the shared Internet. The endpoints of a VPN are connected using abstractions such as Virtual Channels (VCs) of ATM or Label Switching Paths (LSPs) of MPLS technologies. Reliability of an endtoend VPN connection depends on the reliability of the links and nodes in the fixed path that it traverses in the network. In order to ensure service quality and availability in a VPN, seamless recovery from failures is essential. This work considers the problem of fast recovery in the recently proposed VPN hose model. In the hose model bandwidth is reserved for traffic aggregates instead of pairwise specifications to allow any traffic pattern among the VPN endpoints. This work assumes that the VPN endpoints are connected using a tree structure and at any time, at most one tree link can fail (i.e., single link failure model). A restoration algorithm must select asetofbackup edges and allocate necessary bandwidth on them in advance, so that the traffic disrupted by failure of a primary edge can be rerouted via backup paths. We aim at designing an optimal restoration algorithm to minimize the total bandwidth reserved on the backup edges. This problem is a variant of optimal graph augmentation problem which is NPComplete. Thus, we present a polynomialtime approximation algorithm that guarantees a solution which is at most 16 times of the optimum. The algorithm is based on designing two reductions to convert the original problem to one of adding minimum cost edges to the VPN tree so that the resulting graph is 2connected, which can be solved in polynomial time using known algorithms. The two reductions introduce approximation factors of 8 and 2, respectively, thus resulting in a 16appro...