On the minimum local-vertex-connectivity augmentation in graphs

Tight Approximation Algorithm for Connectivity Augmentation Problems

“…As was mentioned, undirected edge-CA is in P [7]. The node-CA (and the element-CA) turned to be NP-hard even when the input graph G 0 is connected and r(u, v) ∈ {0, 2} (c.f., [19]). However, while the element-CA admits a 7/4-approximation algorithm [20], the undirected node-CA with r(u, v) ∈ {0, k} cannot be approximated within O(2 log 1−ε n ) for any fixed ε > 0, unless NP ⊆ DTIME(n polylog(n) ), see [20].…”

“…For more work on CA problems see, e.g., [1,7,10,13,19,21,20], and surveys in [7][8][9]23]. For work on other types of GSN costs see detailed surveys in [15,16] on known upper and lower bounds with respect to approximation.…”

“…Previous work on CA problems that does not follow from results for GSN dealt mainly with algorithm for some special cases, for which were given either polynomial algorithm (c.f., [24,7,10,8]), or constant ratio approximation algorithms (c.f., [12,13,19,17,21,20]). Our main result resolves the approximability of a fundamental case: directed CA with {0, 1}-requirements, thus showing that the approximation threshold Ω(log n) established by A. Frank [7] in the 90's is achievable.…”

A 4/3-approximation for the minimum 2-local-vertex-connectivity augmentation in a connected graph

“…Jordán [3] first proved the NP-hardness of the LVCAP in the case where a given graph G is |V |/2-vertex-connected and a target function r satisfies r(u, v) ∈ {0, |V |/2+1}, u, v ∈ V . Recently, H. Nagamochi and T. Ishii [5] showed that the LVCAP is NP-hard even if G is (k − 1)-vertex-connected and r(u, v) ∈ {0, k}, u, v ∈ V holds for any constant k 2.…”

Approximating Minimum Cost Connectivity Problems via Uncrossable Bifamilies and Spider-Cover Decompositions

“…For r(u, v) ∈ {0, 2} the problem is NP-hard and admits a 3/2-approximation algorithm [43]. For uniform requirements r(u, v) = k for all u, v ∈ V the complexity status is not known for undirected graphs, but for any fixed k an optimal solution can be computed in polynomial time [23].…”