Hardness of k-Vertex-Connected Subgraph Augmentation Problem
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Abstract
Given a k-connected graph G=(V,E) and V′⊂V, k-Vertex-Connected Subgraph Augmentation Problem (k-VCSAP) is to find S⊂V∖V′ with minimum cardinality such that the subgraph induced by V′∪S is k-connected. In this paper, we study the hardness of k-VCSAP in undirect graphs. We first prove k-VCSAP is APX-hard. Then, we improve the lower bound in two ways by relying on different assumptions. That is, we prove no algorithm for k-VCSAP has a PR better than O(log (log n)) unless P=NP and O(log n) unless NP⊆DTIME(nO(log log n)), where n is the size of an input graph.
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Network survivability Graph connectivityPreview
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References
- Akyildiz IF, Su W, Sankarasubramaniam Y, Cayirci E (2002) A survey on sensor networks. IEEE Commun Mag 40:102–114 CrossRefGoogle Scholar
- Feige U, Halldórsson MM, Kortsarz G, Srinivasan A (2002) Approximating the domatic number. SIAM J Comput 32(1):172–195 CrossRefMathSciNetMATHGoogle Scholar
- Frederickson GN, JáJá J (1981) Approximation algorithms for several graph augmentation problems. SIAM J Comput 10(2):270–283 CrossRefMathSciNetMATHGoogle Scholar
- Johnson DS (1973) Approximation algorithms for combinatorial problems. In: Proc of the fifth annual ACM symposium on theory of computing, pp 38–49 Google Scholar
- Jordan T (1997) A note on the vertex-connectivity augmentation problem. University of Southern Denmark Google Scholar
- Kann V (1991) Maximum bounded 3-dimensional matching is MAX SNP-complete. Inf Process Lett 37:27–35 CrossRefMathSciNetMATHGoogle Scholar
- Kortsarz G, Krauthgamer R, Lee JR (2004) Hardness of approximation for vertex-connectivity network design problems. SIAM J Comput 33(3):704–720 CrossRefMathSciNetMATHGoogle Scholar
- Lau LC, Naor JS, Salavatipour MR, Singh M (2007) Survivable network design with degree or order constraints. In: Proceedings of the thirty-ninth annual ACM symposium on theory of computing, June 11–13, 2007, San Diego, California, USA Google Scholar
- Petrank E (1992) The hardness of approximation: gap location. Technical report 754, Computer Science Department, Technion, Israel Institute of Technology, Haifa, Israel Google Scholar
- Raz R, Safra S (1997) A sub-constant error-probability low-degree test, and sub-constant error-probability PCP characterization of NP. In: Proc of the twenty-ninth annual ACM symposium on theory of computing, pp 475–484 Google Scholar
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