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(n O(log log n)), where n is the size of an input graph.
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This work is supported in part by the National Scientific Foundation under Grant Nos. IIS-0513669, CCF-0750992, and CCF-0621829. This work is also supported in part by the National Natural Science Foundation of China Grant Nos. 60553001 and 60604033, the National Basic Research Program of China Grant Nos. 2007CB807900, 2007CB807901, and the Hi-Tech research & Development Program of China Grant No. 2006AA10Z216.
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Ma, C., Kim, D., Wang, Y. et al. Hardness of k-Vertex-Connected Subgraph Augmentation Problem. J Comb Optim 20, 249–258 (2010). https://doi.org/10.1007/s10878-008-9206-5
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DOI: https://doi.org/10.1007/s10878-008-9206-5