Maximum bounded 3-dimensional matching is MAX SNP-complete
Information Processing Letters
Augmenting graphs to meet edge-connectivity requirements
SIAM Journal on Discrete Mathematics
Minimal edge-coverings of pairs of sets
Journal of Combinatorial Theory Series B
Connectivity and network flows
Handbook of combinatorics (vol. 1)
Handbook of combinatorics (vol. 1)
Approximation algorithms for finding highly connected subgraphs
Approximation algorithms for NP-hard problems
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
SIAM Journal on Computing
On the minimum local-vertex-connectivity augmentation in graphs
Discrete Applied Mathematics
Hardness of Approximation for Vertex-Connectivity Network Design Problems
SIAM Journal on Computing
Approximating connectivity augmentation problems
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Independence free graphs and vertex connectivity augmentation
Journal of Combinatorial Theory Series B
Iterative rounding 2-approximation algorithms for minimum-cost vertex connectivity problems
Journal of Computer and System Sciences - Special issue on FOCS 2001
Approximation algorithms for the Label-CoverMAX and Red-Blue Set Cover problems
Journal of Discrete Algorithms
Tight approximation algorithm for connectivity augmentation problems
Journal of Computer and System Sciences
Inapproximability of survivable networks
Theoretical Computer Science
Approximating Steiner Networks with Node-Weights
SIAM Journal on Computing
Approximating minimum-cost connectivity problems via uncrossable bifamilies
ACM Transactions on Algorithms (TALG)
An overview of algorithms for network survivability
ISRN Communications and Networking
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Let G = (V,E) be an undirected graph and let S ⊆ V. The S-connectivity λSG(u,v) of a node pair (u,v) in G is the maximum number of uv-paths that no two of them have an edge or a node in S − {u,v} in common. The corresponding Connectivity Augmentation (CA) problem is: given a graph G = (V,E), a node subset S ⊆ V, and a nonnegative integer requirement function r(u,v) on V × V, add a minimum size set F of new edges to G so that λSG+F(u,v) ≥ r(u,v) for all (u,v) ∈ V × V. Three extensively studied particular cases are: the Edge-CA (S = ∅), the Node-CA (S = V), and the Element-CA (r(u,v)= 0 whenever u ∈ S or v ∈ S). A polynomial-time algorithm for Edge-CA was developed by Frank. In this article we consider the Element-CA and the Node-CA, that are NP-hard even for r(u,v) ∈ {0,2}. The best known ratios for these problems were: 2 for Element-CA and O(rmax ⋅ ln n) for Node-CA, where rmax = maxu,v ∈ V r(u,v) and n = |V|. Our main result is a 7/4-approximation algorithm for the Element-CA, improving the previously best known 2-approximation. For Element-CA with r(u,v) ∈ {0,1,2} we give a 3/2-approximation algorithm. These approximation ratios are based on a new splitting-off theorem, which implies an improved lower bound on the number of edges needed to cover a skew-supermodular set function. For Node-CA we establish the following approximation threshold: Node-CA with r(u,v) ∈ {0,k} cannot be approximated within O(2log1−&epsis; n) for any fixed &epsis; 0, unless NP ⊆ DTIME(npolylog(n)).