S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. In Proceedings of the 27th Annual ACM Symposium on Theory of Computing, pages 1--10, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....factor ff of the optimum value. The quantity ff is called the approximation guarantee of the algorithm. Previous results Results in this paper Undirected Graphs Digraphs Undirected Graphs Digraphs k ECSS 2 [1=k]fork 2[K96] 1.61 for k = 1 [KRY96] 1 [2= k 1) 1 [4= p k] 1. 8 for k 2 [KR 96] 2 for k O(log n) k [Ka 94] k NCSS 1.5 for k = 2 [GSS 93] 1.61 for k = 1 [KRY96] 1 [1=k] 1 [1=k] 2 for k 3 2 for k Table 1: A summary of previous new approximation guarantees for minimum size k edge connected spanning subgraphs (k ECSS) and minimum size k node connected spanning ....

....2 NCSS problem. For graphs and the general minimum size k ECSS problem, first Karger [Ka 94] used randomized rounding to improve the approximation guarantee (for k large w.r.t. log n)to1 [O(log n) k]# Karger s algorithm is not deterministic but Las Vegas. Then Khuller Raghavachari [KR 96] improved the approximation guarantee (for all k) from 2 to (roughly) 1:85. They left open the problem of improving on the approximation guarantee of two for the minimum size k NCSS problem. For digraphs and the problem of finding a minimum size 1 connected (i.e. strongly connected) spanning ....

S. Khuller and B. Raghavachari, "Improved approximation algorithms for uniform connectivity problems," Journal of Algorithms 21 (1996), 434--450. Preliminary version in: Proc. 27th Annual ACM STOC 1995, 1--10.

.... for nding a minimum cost k outconnected subdigraph of a digraph (directed graph) and (2) an upper bound on the order of 3 critical graphs by Mader [17] The Frank Tardos algorithm has been applied earlier to the mincost k VCSS problem by several authors, starting with Khuller and Raghavachari [15]; see also [1, 2, 16] Our second algorithm works for both undirected and directed graphs and is based on a structural charachterization of basic solutions of the relaxation and iterative rounding. We introduce a special class of requirement functions f , namely, skew bisupermodular functions ....

S.Khuller and B.Raghavachari, Improved approximation algorithms for uniform connectivity problems, Journal of Algorithms, 21, 1996, pp. 434-450.

....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. when the costs satisfy the triangle inequality) an approximation ratio 2 (k 1) is given in [KN00] building on a ratio 2 2(k 1) previously shown in [KR96] For uniform costs, an ....

.... 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. when the costs satisfy the triangle inequality) an approximation ratio 2 (k 1) is given in [KN00] building on a ratio 2 2(k 1) previously shown in [KR96] For uniform costs, an approximation ratio of 1 1 k is obtained in [CT00] For a complete Euclidean graph, a polynomial time approximation scheme (i.e. factor 1 # for any fixed # 0) is devised in [CL99] The connectivity augmentation problem has also attracted a lot of attention. A ....

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S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. J. Algorithms, 21(2):434--450, 1996.

....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 k = 2, ADNP99] for k = 2; 3, DN99] for k = 4; 5, and [KN00] for k = 6; 7) For metric costs (i.e. when the costs satisfy the triangle inequality) an approximation ratio 2 (k 1) is given in [KN00] building on a ratio 2 2(k 1) previously shown in [KR96] For uniform costs, an ....

.... d(k 1) 2e 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. when the costs satisfy the triangle inequality) an approximation ratio 2 (k 1) is given in [KN00] building on a ratio 2 2(k 1) previously shown in [KR96] For uniform costs, an approximation ratio of 1 1=k is obtained in [CT00] For a complete Euclidean graph, a polynomial time approximation scheme (i.e. factor 1 for any xed 0) is devised in [CL99] The connectivity augmentation problem has also attracted a lot of attention. A ....

[Article contains additional citation context not shown here]

S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. J. Algorithms, 21(2):434-450, 1996.

....algorithm was improved by Khuller and Thurimella [9] with regards to the time complexity. Khuller and Vishkin [10] proposed a similar approach for the related problem of identifying a minimum weight edge biconnected spanning subgraph when no starting graph G 0 is given. Khuller and Raghavachari [12] proposed another algorithm using similar basic ideas for the problem of identifying a (nonspanning) subgraph with a given edge or vertex connectivity for a given graph G. A survey on several related problems and approximation algorithms is given by Khuller [11] Algorithms with better worst case ....

Khuller S., Raghavachari B.: Improved Approximation Algorithms for Uniform Connectivity Problem, Journal of Algorithms 21(2), (1996), 434--450

....than a k approximation. For the more specialized connectivity problems of constructing a minimum cost uniformly k connected graph, the best known approximation guarantee is roughly factor k [3, 13] 1 If c is a metric, then there are constant factor approx imations for uniform k connectivity [12, 13]. Last IPCO, Melkonian and Tardos [14] extend Jain s iterative rounding analysis to obtain a 4 approximation for uniform k edge connectivity on directed graphs. They also describe a different approach that yields a 2 approximation. There are numerous approximation results for other special cases ....

S. Khuller and B. Raghavachaxi. Improved approximation algorithms for uniform connectivity problems. J. Algorithms, 1996.

....algorithm for VC SNDP is currently known. However, approximation algorithms are known in special cases. In the case 8) S B T , there is a approximation algorithm due to Kortsarz and Nutov [14] furthermore, when 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, ....

S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. J. Algorithms, 21:434--450, 1997.

.... Enomoto, Iida, and Ota [EIO96] investigate spanning subgraphs in which every vertex has a degree in some interval. Barnette [Bar94] Gao [Gao95] and Sanders and Zhao [SZ96] consider biconnected spanning subgraphs with bounded degree. Khuller and Vishkin [KV94] Khuller and Raghavachari [KR95] Cheriyan and Thurimella [CT96a] and Khuller [Khu97] approximate the smallest and construct a small k (edge) connected spanning subgraph. Khuller and Raghavachari [KR95] Khuller, Raghavachari, and Young [KRY96] and Goemans and Williamson [GW96] present results for the more general ....

....[SZ96] consider biconnected spanning subgraphs with bounded degree. Khuller and Vishkin [KV94] Khuller and Raghavachari [KR95] Cheriyan and Thurimella [CT96a] and Khuller [Khu97] approximate the smallest and construct a small k (edge) connected spanning subgraph. Khuller and Raghavachari [KR95] Khuller, Raghavachari, and Young [KRY96] and Goemans and Williamson [GW96] present results for the more general edgeweighted case. The papers of Camerini, Galbiati, and Maffioli [CGM80] and Raghavachari [Rag97] survey related results in this area. As stated we are mainly interested in the ....

S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. In 27th Annual ACM Symposium on Theory of Computing (STOC`95), pages 1--10, 1995.

.... 8) 21 Edge and Vertex Connectivity: Given a weighted graph G = V; E) and an integer k, find a minimum weight edge 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, ....

Khuller, S. and Raghavachari, B. (1996). Improved approximation algorithms for uniform connectivity problems. Journal of Algorithms, 21(2):434--450.

.... (that is, when the weight function satisfies the triangle inequality) and uniform weights (that is, when the weight of every edge is the same) and there has been extensive recent research on approximation algorithms for this and related problems with uniform weights and with metric weights, see [1, 3, 7, 9, 10]. For metric weights, Khuller and Raghavachari [10] developed a (2 2(k Gamma 1) n) approximation algorithm for the minimum weight k node connected subgraph problem. For uniform weights, Cheriyan and Thurimella [3] gave a (1 1 k ) approximation algorithm. We design approximation algorithms ....

.... and uniform weights (that is, when the weight of every edge is the same) and there has been extensive recent research on approximation algorithms for this and related problems with uniform weights and with metric weights, see [1, 3, 7, 9, 10] For metric weights, Khuller and Raghavachari [10] developed a (2 2(k Gamma 1) n) approximation algorithm for the minimum weight k node connected subgraph problem. For uniform weights, Cheriyan and Thurimella [3] gave a (1 1 k ) approximation algorithm. We design approximation algorithms for the following problem that is sandwiched between ....

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S.Khuller and B.Raghavachari, "Improved approximation algorithms for uniform connectivity problems," Journal of Algorithms 21 (1996), 434--450.

.... this basic algorithmic outline leads to the desired approximation factors were also neatly summarized in [9] For the latter problem, also known as the minimum cost k vertex connected subgraph problem, there is also a (2 2(k Gamma1) n ) approximation algorithm due to Khuller and Raghavachari [6] in the case that edge costs obey the triangle inequality. However, no non trivial approximation algorithm is known for the vertex connectivity survivable network design problem in its full generality. In this paper we make progress on this important problem by considering a natural problem ....

S. Khuller and B. Raghavachari, Improved Approximation Algorithms for Uniform Connectivity Problems, STOC '95, pp. 1--10.

....it connected. Any Hamiltonian cycle is 2 node connected. Thus, the length of a minimum cost 2 node connected subgraph is a lower bound on opt. Notice that it is NP hard to find the length of the minimum cost 2 node connected subgraph. However, a 2 approximation can be computed in polynomial time [8, 15]. Let S 2 denote the square of graph S ( u; v) is an edge in S 2 if there is a path in S between u and v using at most 2 edges) We first provide an 8 approximation and then show how it can be improved to obtain a 4 approximation. The 8 approximation algorithm is described below. 1. Find an ....

S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. Journal of Algorithms, 21:434--450, 1996.

....node from S leaves it connected. Any Hamiltonian cycle is 2 node connected. Thus, the minimum weight 2 node connected subgraph is a lower bound on opt. Notice that it is np hard to find the minimum weight 2 node connected subgraph. However, a 2 approximation can be computed in polynomial time [10, 18]. Let S 2 denote the square of graph S. Thus, u; v) is an edge in S 2 if there is a path in S between u and v using at most 2 edges. Our algorithm is described below. tsp approx (G) 1. Find an (approximately) minimum cost 2 node connected subgraph of G. Call this graph S. 2. Find a ....

S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. Journal of Algorithms, 21:434--450, 1996.

....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 ....

....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: fauletta,parenteg dia.unisa.it. 2 an approximation algorithm that obtains a ....

[Article contains additional citation context not shown here]

S. Khuller, B. Raghavachari, Improved Approximation Algorithms for Uniform Connectivity Problems, J. of Algorithms 21, 434-450, 1996.

....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 algorithm due to Khuller and Raghavachari [6] in the case that edge costs obey the triangle inequality. However, no non trivial approximation algorithm is known for the vertex connectivity survivable network design problem in its full generality. In this paper we make progress on this important problem by considering a natural problem ....

S. Khuller and B. Raghavachari, Improved Approximation Algorithms for Uniform Connectivity Problems, STOC '95, pp. 1-10.

....spanning subgraph (kECSS) is known to be NP hard [10] Khuller and Vishkin [16] obtained a 2 approximation algorithm for weighted k ECSS for k 1. Nagamochi and Ibaraki [18] gave an ecient algorithm for nding a sparse k connected subgraph of a given graph. Khuller and Raghavachari [15] demonstrated an algorithm for k ECSS with approximation ratio of 1.85 and an algorithm for k VCSS with a performance ratio of 2 n in graphs satisfying the triangle inequality. Fernandes [6] improved the approximation ratio to 1.75 for k ECSS. Cheriyan and Thurimella [3] presented an elegant ....

....on multigraphs as well, whereas the CT algorithm guarantees a ratio of 3 2 for 3 ECSS only on simple graphs. In fact Gabow [9] shows examples of multigraphs for which the ratio obtained by the CT algorithm is arbitrarily close to 2. In that paper he also shows how to modify an earlier algorithm of [15] for k ECSS to multigraphs with a performance ratio of about 1.61. 2 De nitions Let G = V; E) be the given graph with jV j = n. Let Opt be an optimal 2 ECSS of G. We will also use Opt to denote the cardinality of an optimal 2 ECSS of G, and this should cause no confusion. A node v is called a ....

S. Khuller and B. Raghavachari, Improved approximation algorithms for uniform connectivity problems, J. Algorithms, 21, pp. 433-450, 1996.

....accurate approximation ratio when compared to the previous best 3 2 approximation ratio obtained by Cheriyan and Thurimella [2] and Gabow [3] for this problem, thereyby showing that the previous ratio of 3=2 can certainly be improved. Khuller and Raghavachari gave a 1. 85 approximation algorithm [7] for k ECSS problem which was later simpli ed [6] to obtain a 2 1=k approximation ratio. For 3 ECSS their algorithm gives a 5 3 ratio. Cheriyan and Thurimella [2] presented an improved approximation algorithm for k ECSS with 1 2= k 1) ratio for simple graphs. For k = 3, their ratio is 3=2. ....

....and Thurimella [2] presented an improved approximation algorithm for k ECSS with 1 2= k 1) ratio for simple graphs. For k = 3, their ratio is 3=2. For multigraphs, Gabow [3] presented a 3=2 approximation algorithm for 3 ECSS. He also improved the analysis of the algorithm presented in [7] for k ECSS problem, resulting in an improved approximation ratio of strictly less than 1.61 for arbitrary multigraphs. For the 2 ECSS problem, Khuller and Vishkin presented a 3 2 approximation algorithm using simple DFS approach. Developing on their work, Cheriyan, Seb o and Szigeti [1] improved ....

S. Khuller and B. Raghavachari, Improved approximation algorithms for uniform connectivity problems, J. Algorithms, 21, pp. 433-450, 1996.

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S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. In Proceedings of the 27th Annual ACM Symposium on Theory of Computing, pages 1--10, 1995.

No context found.

S. Khuller and B. Raghavachari, Improved approximation algorithms for uniform connectivity problems, J. Algebra, 21 (1996), pp. 434--450.

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S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. J. Algorithms, 21(2):434--450, 1996.

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S. Khuller and B. Raghavachari, Improved Approximation Algorithms for Uniform Connectivity Problems, Journal of Algorithms 21 (1996), pp. 434--450.

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S. Khuller and B. Raghavachari, Improved approximation algorithms for uniform connectivity problems, J. Algorithms, 21 (1996), pp. 434--450.

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S. Khuller and B. Raghavachari, "Improved Approximation Algorithms for Uniform Connectivity Problems," Journal of Algorithms, 21, 434--450, 1996.

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S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. J. Algorithms, 21:434--450, 1997.

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S. Khuller and B. Raghavachari, Improved approximation algorithms for uniform connectivity problems, Journal of Algorithms 21, pp. 433-450, 1996.

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