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34
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 82 (3 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...
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 69 (2 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 59 (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 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 50 (6 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...
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 40 (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.
A 2approximation algorithm for finding an optimum 3vertexconnected spanning subgraph
 Journal of Algorithms
, 1999
"... The problem of finding a minimum weight kvertex connected spanning subgraph in a graph G = (V, E) is considered. For k ≥ 2, this problem is known to be NPhard. Combining properties of inclusionminimal kvertex connected graphs and of koutconnected graphs (i.e., graphs which contain a vertex from ..."
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Cited by 33 (15 self)
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The problem of finding a minimum weight kvertex connected spanning subgraph in a graph G = (V, E) is considered. For k ≥ 2, this problem is known to be NPhard. Combining properties of inclusionminimal kvertex connected graphs and of koutconnected graphs (i.e., graphs which contain a vertex from which there exist k internally vertexdisjoint paths to every other vertex), we derive an auxiliary polynomial time algorithm for finding a ( ⌈ k ⌉ + 1)connected subgraph 2 with a weight at most twice the optimum to the original problem. In particular, we obtain a 2approximation algorithm for the case k = 3 of our problem. This improves the best previously known approximation ratio 3. The complexity of the algorithm is O(V  3 E) = O(V  5).
When cycles collapse: A general approximation technique for constrained twoconnectivity problems
, 1993
"... 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(n 2 log n) time on a graph with n nodes and ..."
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Cited by 31 (9 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(n 2 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 ...
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 23 (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...
A Better Approximation Ratio for the Minimum Size kEdgeConnected Spanning Subgraph Problem
, 1998
"... Consider the minimum size kedgeconnected spanning subgraph problem: given a positive integer k and a kedgeconnected graph G, nd a kedgeconnected spanning subgraph of G with the minimum number of edges. This problem is known to be NPcomplete. Khuller and Raghavachari presented the rst algorit ..."
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Cited by 15 (0 self)
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Consider the minimum size kedgeconnected spanning subgraph problem: given a positive integer k and a kedgeconnected graph G, nd a kedgeconnected spanning subgraph of G with the minimum number of edges. This problem is known to be NPcomplete. Khuller and Raghavachari presented the rst algorithm which, for all k, achieves a performance ratio smaller than a constant which is less than two. They proved an upper bound of 1.85 for the performance ratio of their algorithm. Currently, the best known performance ratio for the problem is 1+2=(k+ 1), achieved by a slower algorithm of Cheriyan and Thurimella. In this paper, we improve Khuller and Raghavachari's analysis, proving that the performance ratio of their algorithm is smaller than 1.7 for large enough k, and that it is at most 1.75 for all k. Second, we show that the minimum size 2edgeconnected spanning subgraph problem is MAX SNPhard. A preliminary version of this paper appeared in the Proceedings of the 8 th Annual ACM...