C. G. Fernandes. A better approximation ratio for the minimum size k-edge-connected spanning subgraph problem. Journal of Algorithms, 28:105--124, 1988.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....on k. We would prefer a polynomial time approximation scheme (PTAS) an algorithm taking an instance G and a positive #, returning a solution with cost at most (1 #) times optimal, and running in time that is polynomial for each fixed #. However, even for k = 2 these problems are MaxSNP hard [5, 7], so there is no PTAS unless P=NP. # This work was supported by NSF Grant number CCR 9820931. Department of Mathematics and Computer Science, Emory University, Atlanta GA 30322, USA. Email: mic mathcs.emory.edu. Same support and address. Email: psissok emory.edu. However, we may still find a ....

C. G. Fernandes. A Better Approximation Ratio for the Minimum Size k-edge-Connected Spanning Subgraph Problem. Journal of Algorithms, 28:105--124, 1988.

....with uniform costs is strictly easier to approximate than k VCSS with edge costs 0 and 1, unless P = NP. Finally, the analysis in Theorem 2 and the preceding corollary extend also to the corresponding edge connectivity problem. Such a hardness of approximation result was previously shown in [Fer98] but our proof is considerably simpler. Theorem 5. For some fixed # 0, it is NP hard to approximate the minimumcost 2 edge connected spanning subgraph (2 ECSS) problem with uniform costs within a factor of (1 #) 2.2 The vertex connectivity augmentation problem The vertex connectivity ....

C. G. Fernandes. A better approximation ratio for the minimum size k- edge-connected spanning subgraph problem. J. Algorithms, 28(1):105--124, 1998.

....with uniform costs is strictly easier to approximate than k VCSS with edge costs 0 and 1, unless P = NP. Finally, the analysis in Theorem 2.1 and the preceding corollary extend also to the corresponding edge connectivity problem. Such a hardness of approximation result was previously shown in [Fer98] but our proof is considerably simpler. Theorem 2.6. For some xed 0, it is NP hard to approximate the minimum cost 2 edgeconnected spanning subgraph (2 ECSS) problem with uniform costs within a factor of 1 . 2.2 The vertex connectivity augmentation problem The vertex connectivity ....

C. G. Fernandes. A better approximation ratio for the minimum size k-edge-connected spanning subgraph problem. J. Algorithms, 28(1):105-124, 1998.

....to all of the considered problems. That is why we assume c Previous results. 2 EC 2 VC: This is the simplest non trivial version of the connectivity problem and has been studied for a long time, but tight approximation guarantees and inapproximability results are not fully understood yet [7, 17, 25, 33, 29, 13, 8]. For 2 EC, Khuller and Vishkin [25] gave a 2 approximation, improved by Cheriyan et al. 7] to 12 , and to 3 by Vempala and Vetta [33] The best known result, due to Krysta and Kumar [29] is 1344 ) approximation. For 2 VC, Khuller and Vishkin [25] gave a approximation, improved to by ....

....The best known result, due to Krysta and Kumar [29] is 1344 ) approximation. For 2 VC, Khuller and Vishkin [25] gave a approximation, improved to by Garg et al. 17] and to by Vempala and Vetta [33] Both 2 VC and 2 EC problems are NP hard even on 3 regular planar graphs. Fernandes [13] proved Max SNP hardness on arbitrary graphs; Czumaj Lingas [8] show Max SNP hardness on bounded degree 6 graphs. These results do not give explicit hardness constants. TSP(1,2) For this version of the TSP, Karp has shown NP completeness in his seminal paper [22] Papadimitriou and ....

C.G. Fernandes. A better approximation ratio for the minimum size k-edge-connected spanning subgraph problem. J. Algorithms, 28, 105-124,

....the connectivity problems. Our further results concern hardness of approximation for the minimum cost k vertex and k edge connected spanning subgraph problem. To the best of our knowledge (cf. also [4] the only known inapproximability result for the connectivity problems is due to Fernandes [6]. She shows that the k edge connectivity problem in unweighted graphs is MaxSNP hard even for k = 2: By the characterization of Max SNP provided in [2] this result implies that the minimum cost k edge connected spanning subgraph problem does not have PTAS unless P = NP: In this paper we add ....

....Inapproximability for k edge connectivity One can modify all the preceding proofs to obtain similar inapproximability results for the problem of finding a minimum cost k edge connected subgraph of a k vertex connected graph. A theorem similar to Theorem 5. 6 has been already proven by Fernandes [6], but we believe that our proof is more intuitive and significantly simpler. Theorem 5.4. There exist constants Delta 0 0 and 0 such that, given an 1 2 Delta 0 graph G on n vertices, and given the promise that either its minimum weight 2 edge connected spanning subgraph H is of weight ....

C. G. Fernandes. A better approximation ratio for the minimum size k-edge-connected spanning subgraph problem. J. Algorithms, 28(1):105--124, July 1998.

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C. G. Fernandes. A better approximation ratio for the minimum size k-edge-connected spanning subgraph problem. Journal of Algorithms, 28:105--124, 1988.

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C. G. Fernandes, A better approximation ratio for the minimum size k-edge-connected spanning subgraph problem, J. Algorithm, 28 (1998), pp. 105--124.

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C. G. Fernandes. A better approximation ratio for the minimum size k-edge-connected spanning subgraph problem. J. Algorithms, 28(1):105--124, 1998.

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