R. Ravi and D. P. Williamson. An approximation algorithm for minimum-cost vertexconnectivity problems. In Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 332--341, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the edge node connectivityby adding a minimum number of new edges (each new edge costs one) For instance, given a tree, one maywant to add the minimum number of new edges to achieve 3 node connectivity. Readers interested in network design with arbitrary edge costs are referred to [GW 95]and[RW 95] and readers interested in edge node connectivity augmentation problems for both graphs and directed graphs are referred to [F 94] Let us focus on node connectivity augmentation problems: given a graph, increase the node connectivitytok by adding the minimum number of new edges. The case ....

R. Ravi and D. P. Williamson, "An approximation algorithm for minimum-cost vertexconnectivity problems," Preliminary version in Proc. 6 ACM-SIAM S.O.D.A.(O[D[A 332-341. To appear in Algorithmica.

....It follows that this is an upper bound on the integrality ratio for the linear programming relaxation. The previous best approximation guarantee was more than k=2 [16] and the previous best upper bound on the integrality ratio was (k) An O(log k) approximation guarantee was claimed earlier in [20], but subsequently an error has been found and that claim is withdrawn. This algorithm is based on two results: 1) a polynomial time algorithm of Frank and Tardos [6] for nding a minimum cost k outconnected subdigraph of a digraph (directed graph) and (2) an upper bound on the order of ....

R.Ravi and D.P.Williamson, An approximation algorithm for minimum-cost vertex-connectivity problems. Algorithmica, 18, 1997.

....for the problem. Their first algorithm achieves approximation ratio 6H(k) O(log k) where H(k) is the kth harmonic number, for the case where k n 6. Their second algorithm achieves an approximation ratio n # for the case where k (1 #)n. An approximation ratio of O(log k) claimed in [RW97] was found to be erroneous, see [RW02] Better approximation ratios are known for several special cases of k VCSS. For k 7 an approximation ratio of #(k 1) 2# is known (see [KR96] for k = 2, ADNP99] for k = 2, 3, DN99] for k = 4, 5, and [KN00] for k = 6, 7) For metric costs (i.e. ....

R. Ravi and D. P. Williamson. An approximation algorithm for minimumcost vertex-connectivity problems. Algorithmica, 18(1):21--43, 1997.

....for the problem. Their rst algorithm achieves approximation ratio 6H(k) O(log k) where H(k) is the kth harmonic number, for the case where k n=6. Their second algorithm achieves an approximation ratio n= for the case where k (1 )n. An approximation ratio of O(log k) claimed in [RW97] was found to be erroneous, see [RW02] In [CKK02] improved approximations are given for 2 VCSS in dense graphs and graphs with maximum degree 3. Better approximation ratios are known for several special cases of k VCSS. For k 7 an approximation ratio of d(k 1) 2e is known (see [KR96] for ....

R. Ravi and D. P. Williamson. An approximation algorithm for minimum-cost vertexconnectivity problems. Algorithmica, 18(1):21-43, 1997.

....by Goemans et al. gives a O(log k) approximation [5] This is a primal dual based approximation algorithm that does not rely on solving linear programs. No nontrivial approximation algorithm is known for MCVC. For the prob lem where r is restricted to 0, 1, 2 vXv, Ravi and Williamson [16] describe a primal dual 3 approximation algorithm. We call this problem 0, 1, 2 MCVC. This problem arises in the design of survivable communications networks [8, 15] In this paper we describe a 2 approximation algorithm for 0, 1, 2 MCVC that iteratively rounds appropriately defined linear ....

....subset of vertices that requires that the number of edges leaving the set be at least the maximum connectivity requirement over all pairs of vertices that have exactly one member of the pair in the set. Let f(S) be defined to take this value for subset S. Let 5(S) denote the set of edges with 1 In [16], a primal dual algorithm is proposed, but the analysis has recently discov ered to be flawed, and the algorithm does not provide the claimed O(log k) factor guarantee [17] exactly one endpoint in . The linear program is: s.t. Y] ees(s) x(e) f(S) VS C V (MCEC) 0 x(e) 1, Ve E The ....

R. Ravi and D. P. Williamson. An approximation algorithm for minimum-cost vertex-connectivity problems. Algorithmica, 18(1):21-43, 1997.

....set E 0 E such that between any pair of vertices, there are k edge disjoint paths in the graph G 0 = V; E 0 ) Frederickson and J aJ a, 1981; Khuller and Vishkin, 1994; Khuller and Raghavachari, 1996; Hochbaum, 1995, Ch. 6) Similar algorithms handle the goal of k vertex disjoint paths (Ravi and Williamson, 1995; Frederickson and J aJ a, 1981; Khuller and Vishkin, 1994; Garg et al. 1993) and the goal of augmenting a given graph to achieve a given connectivity (Frederickson and J aJ a, 1981; Khuller and Thurimella, 1993) Steiner Tree: Given an undirected graph with positive edge weights and a subset ....

Ravi, R. and Williamson, D. P. (1995). An approximation algorithm for minimum-cost vertexconnectivity problems. In (ACM/SIAM, 1995), pages 332--341.

....connectivity degree can be seen as a measure of the tolerance to failures occurring in the nodes of the network. For more motivations see e.g. 8] The k MWVC problem is NP hard, for k 2, even in presence of unweighted edges. Various approximation algorithms for its solution have been proposed in [10, 14, 4]. An approximation algorithm with factor ff, is a polynomial time algorithm that returns a solution whose cost is within a factor ff from the cost of an optimal solution. The best approximation algorithm for the k MWVC problem is due to Ravi and Williamson [14] and it obtains a factor 2H(k) where ....

....solution have been proposed in [10, 14, 4] An approximation algorithm with factor ff, is a polynomial time algorithm that returns a solution whose cost is within a factor ff from the cost of an optimal solution. The best approximation algorithm for the k MWVC problem is due to Ravi and Williamson [14] and it obtains a factor 2H(k) where H(k) P k i=1 1=i. In [10] it is given A Preliminary version of this paper has been presented at STACS 97, see [1] y Dipartimento di Informatica ed Applicazioni, R.M. Capocelli , Universit a di Salerno, 84081 Baronissi (Italy) e mail: ....

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R. Ravi, D.P. Williamson, An Approximation Algorithm for Minimum-Cost Vertex-Connectivity Problems, in Proc. of SODA '95, 332--341, (1995).

....before this can be done. 3 Vertex Connectivity Problems 3.1 Weighted Vertex Connectivity For the general problem no constant factor approximation algorithms are known. The best known algorithm to find a connected subgraph for the weighted case is the algorithm due to Ravi and Williamson [35] that achieves a factor of 2H( where H( 1 1 2 : 1 . For the case of finding a 2 vertex connected graph, an approximation algorithm achieving a factor of 3 was given by Frederickson and J aJ a, through solving the more general graph augmentation problem. It is possible to obtain ....

R. Ravi and D. Williamson, An approximation algorithm for minimum-cost vertexconnectivity problems, to appear in Proc. 6th Annual ACM-SIAM Symposium on Discrete Algorithms, (1995).

....new structural results to obtain improved approximation guarantees. A basic problem in network design is to find a min cost k connected spanning subgraph. In this paper k connectivity refers to k node connectivity. The problem is NP hard, and there is an O(logk) approximation algorithm due to [13]. A generalization is to find a min cost subgraph that has at least k vw openly disjoint paths between every node pair v; w, where [k vw ] is a prespecified connectivity requirements matrix. No poly logarithmic approximation algorithm for this problem is known. For the special case when each k ....

....is to find a min cost subgraph that has at least k vw openly disjoint paths between every node pair v; w, where [k vw ] is a prespecified connectivity requirements matrix. No poly logarithmic approximation algorithm for this problem is known. For the special case when each k vw is in f0; 1; 2g, [13] gives a 3 approximation algorithm. The min cost k outconnected subgraph problem is sandwiched between the basic problem and the general problem. There are two versions of the problem. In the single root version, there is a root node r, and the connectivity requirement is k rv = k, for all ....

R. Ravi and D. P. Williamson, "An approximation algorithm for minimum-cost vertex-connectivity problems." Algorithmica (1997) 18: 21-43.

....contract 200975 92 7. 1 Introduction The study of connectivity in graph theory has important applications in the areas of network reliability and network design. Several approximation algorithms have been developed for the problem of nding subgraphs satisfying certain connectivity requirements [2, 4, 6, 7, 8, 9, 12, 13, 16, 18]. In this paper, we concentrate on the minimum size k edge connected spanning subgraph problem: given a positive integer k and a k edge connected graph G, nd a k edge connected spanning subgraph of G with the minimum number of edges. This problem is known to be NP complete [5] even for k = 2: ....

R. Ravi and D. Williamson, \An Approximation Algorithm for Minimum-Cost VertexConnectivity Problems," Proc. 6 th Annual ACM-SIAM Symposium on Discrete Algorithms, 332-341, 1995.

....This matches the previous best bound for the corresponding edge connectivity problem. 1 Introduction Connectivity is fundamental to the study of graphs and graph algorithms. Recently, many approximation algorithms for finding subgraphs that meet given connectivity requirements have been developed [1, 12, 15, 16, 21, 25, 27]. These results provide practical A preliminary draft of this paper appeared in the proceedings of the 27th Annual ACM Symposium on Theory of Computing (STOC) 1995. y Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742. ....

....an approximation factor of 2 was achieved by Khuller and Vishkin [21] Approximation algorithms with constant performance ratios (for all k) are not known for the k vertex connected subgraph problem. The best known algorithm to find a k vertex connected subgraph is due to Ravi and Williamson [25] that achieves a factor of 2H(k) where H(k) is the kth Harmonic number (H(k) 1 1 2 : 1 k ) For the special case of k = 3, Penn and Shasha Krupnik [24] have used our techniques to obtain an approximation factor of 3. Frederickson and J aJ a [9] considered the problem of computing a ....

[Article contains additional citation context not shown here]

R. Ravi and D. P. Williamson, An approximation algorithm for minimum-cost vertex-connectivity problems, Proc. 6th Annual ACM-SIAM Symp. on Discrete Algorithms, San Francisco, CA, pp. 332--341, (1995).

....degree can be seen as the tolerance to faults occurring in the vertices. For more motivations see e.g. 8] The k MWVC problem is NP hard [7] even in presence of unweighted edges or even when the weights can be just 1 or 2. Various approximation algorithms for its solution have been proposed, [9, 12, 5], and their approximation factors are summarized in Figure 1. An approximation algorithm with factor ff, is a polynomial time algorithm that returns a solution whose cost is no greater than ff times the cost of an optimial solution. The best approximation algorithm for the general case is due to ....

....factors are summarized in Figure 1. An approximation algorithm with factor ff, is a polynomial time algorithm that returns a solution whose cost is no greater than ff times the cost of an optimial solution. The best approximation algorithm for the general case is due to Ravi and Williamson [12] and it obtains a factor 2H(k) where H(k) P k i=1 1=i. Better algorithms have been proposed for particular values of k. In [12] and [5] two algorithms are described that obtain a factor 3 for the case k = 2. In [9] Khuller and Raghavachari improved on this result obtaining a factor of 2 ....

[Article contains additional citation context not shown here]

R. Ravi, D.P. Williamson, An Approximation Algorithm for Minimum-Cost VertexConnectivity Problems, in Proc. 6-th Annual ACM-SIAM Symp. on Discrete Algorithms, 332--341, (1995). Pag.

....This matches the previous best bound for the corresponding edge connectivity problem. 1 Introduction Connectivity is fundamental to the study of graphs and graph algorithms. Recently, many approximation algorithms for finding subgraphs that meet given connectivity requirements have been developed [1, 11, 14, 15, 19, 21, 22]. These results provide practical approximation algorithms for NP hard network design problems, via an increased understanding of connectivity properties. In this paper we focus on uniform k connectivity problems. The term connectivity refers to both edge and vertex connectivities, unless ....

....an approximation factor of 2 was achieved by Khuller and Vishkin [19] Approximation algorithms with constant performance ratios (for all k) are not known for the k vertex connected subgraph problem. The best known algorithm to find a k vertex connected subgraph is due to Ravi and Williamson [21] that achieves a factor of 2H(k) where H(k) is the kth Harmonic number (H(k) 1 1 2 : 1 k ) Frederickson and J aJ a [8] considered the problem of computing a minimum weight biconnected spanning subgraph. They gave an approximation algorithm for a more general graph augmentation ....

[Article contains additional citation context not shown here]

R. Ravi and D. P. Williamson, An approximation algorithm for minimum-cost vertexconnectivity problems, Proc. 6th Annual ACM-SIAM Symp. on Discrete Algorithms, pp. 332--341, (1995).

....the edge node connectivity by adding a minimum number of new edges (each new edge costs one) For instance, given a tree, one may want to add the minimum number of new edges to achieve 3 node connectivity. Readers interested in network design with arbitrary edge costs are referred to [GW 95] and [RW 95] and readers interested in edge node connectivity augmentation problems for both graphs and directed graphs are referred to [F 94] Let us focus on node connectivity augmentation problems: given a graph, increase the node connectivity to k 0 by adding the minimum number of new edges. The case ....

R. Ravi and D. P. Williamson, "An approximation algorithm for minimum-cost vertexconnectivity problems," Preliminary version in Proc. 6th ACM-SIAM S.O.D.A. (1995), 332-341. To appear in Algorithmica.

No context found.

R. Ravi and D. P. Williamson. An approximation algorithm for minimum-cost vertexconnectivity problems. In Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 332--341, 1995.

No context found.

R. Ravi and D.P. Williamson, An approximation algorithm for minimum-cost vertexconnectivity problems, Algorithmica 18 (1997), pp. 21--43.

No context found.

R. Ravi and D. P. Williamson. An approximation algorithm for minimum-cost vertex-connectivity problems. Algorithmica, 18(1):21--43, 1997.

....edge costs obey the triangle inequality, Khuller and Ragavachari [13] give a constant approximation algorithm. Very recently, Cheriyan, Vempala, and Vetta [3] announced a U ( for this problem for graphs that contain at least U ) 6 vertices. In the case E# V B 3 B K , Ravi and Williamson [15] give a primal dual 3 approximation algorithm. Recently, Fleischer [7] showed how to extend the algorithm and the proof of Jain for EC SNDP to this special case of VC SNDP, obtaining a 2 approximation algorithm. Our central result is a generalization of Jain s theorem to a new class of functions ....

....AXE CB DYE W K is a weakly two supermodular function, and a polynomial time separation algorithm exists. This result specializes to Jain s result exactly in the is only non zero when W Z [# . In this case, 9LC; CB Z [X = is a weakly supermodular Ravi and Williamson [15] had claimed a ] for the case of general edge costs, but there is an error in their paper; see [16] for details. function. The weakly two supermodular functions are related to bisupermodular functions, which are the negative of bisubmodular functions. Bisubmodular functions appear as ....

R. Ravi and D. P. Williamson. An approximation algorithm for minimum-cost vertex-connectivity problems. Algorithmica, 18(1):21--43, 1997.

....network design problem arises from problems in the telecommunications industry (c.f. 4, 7] and has been studied from many different approaches including polyhedral combinatorics [10, 4] interchange heuristics [8] min max relations [1] in the unweighted case) approximation algorithms [12, 2, 9], and implementations thereof [7] In this paper, we consider approximation algorithms for the SNDP. A ae approximation algorithm for the SNDP runs in polynomial time and finds a solution of value no more than ae times the value of an optimal solution. There appears to be a qualitative difference ....

....of these algorithmic ideas and the primal dual schema, we refer the reader to the survey article [3] or the book [11] In the case of VC SNDP, however, very little is known in terms of approximation algorithms. Using the basic algorithmic outline established in [12, 2] Ravi and Williamson [9] give a 3approximation algorithm in the case that r uv 2 f0; 1; 2g, and a 2H k approximation algorithm in the case that r uv = k for all u; v 2 V . The conditions under which this basic algorithmic outline leads to the desired approximation factors were also neatly summarized in [9] For the ....

[Article contains additional citation context not shown here]

R. Ravi and D.P. Williamson, An approximation algorithm for minimum-cost vertexconnectivity problems, Algorithmica 18 (1997), pp. 21--43.

....network design problem arises from problems in the telecommunications industry (c.f. 4, 7] and has been studied from many different approaches including polyhedral combinatorics [10, 4] interchange heuristics [8] min max relations [1] in the unweighted case) approximation algorithms [11, 2, 9], and implementations thereof [7] In this paper, we consider approximation algorithms for the SNDP. A ae approximation algorithm for the SNDP runs in polynomial time and finds a solution of value no more than ae times the value of an optimal solution. There appears to be a qualitative difference ....

....few years for EC SNDP [11, 2] where H n = 1 1 2 Delta Delta Delta 1 n ln n and k = max u;v ruv. Very recently, Jain gave a 2 approximation algorithm for the problem [5] However, in the case of VC SNDP very little is known in terms of approximation algorithms. Ravi and Williamson [9] show a 3 approximation algorithm in the case that r uv 2 f0; 1; 2g, and a 2H k approximation algorithm in the case that r uv = k for all u; v 2 V . For the latter problem, also known as the minimumcost k vertex connected subgraph problem, there is also a (2 2(k Gamma1) n ) approximation ....

[Article contains additional citation context not shown here]

R. Ravi and D.P. Williamson, An approximation algorithm for minimum-cost vertex-connectivity problems, Algorithmica 18 (1997), pp. 21--43.

....prior to our work. Khuller and Thurimella [11] give a 3 approximation algorithm for the minimum cost 2 vertex connectivity problem. Khuller [10] shows a 2(1 1 n ) approximation algorithm for the same problem in an n vertex graph. After the appearance of an extended abstract of our paper [18], Khuller and Raghavachari [10] gave a (2 2(k Gamma1) n ) approximation algorithm for the minimum cost k vertex connectivity problem for graphs with edge costs that obey the triangle inequality. Our results do not require the triangle inequality assumption. Most of the known results for ....

R. Ravi and D. P. Williamson. An approximation algorithm for minimum-cost vertexconnectivity problems. In Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 332--341, 1995.

No context found.

R. Ravi and D. P. Williamson. An approximation algorithm for minimum-cost vertexconnectivity problems. Algorithmica, 18(1):21--43, 1997.

No context found.

R. Ravi and D. P. Williamson, Erratum: "An approximation algorithm for minimum-cost vertex-connectivity problems," Algorithmica, 34 (2002), pp. 98--107.

No context found.

R. Ravi and D. P. Williamson, An approximation algorithm for minimum-cost vertexconnectivity problems, Algorithmica, 18 (1997), pp. 21--43.

No context found.

R. Ravi and D. P. Williamson, "An approximation algorithm for minimum-cost vertexconnectivity problems," Preliminary version in Proc. 6th ACM-SIAM S.O.D.A. (1995), 332-341. To appear in Algorithmica.

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