Approximating connectivity augmentation problems
Zeev Nutov · 2005
Let G = (V, E) be an undirected graph and let S ⊆ V. The S-connectivity λS G (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 λS G+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 algorithm for Edge-CA was developed by Frank. In this paper 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−ε n) for any fixed ε> 0, unless NP ⊆ DTIME(npolylog(n)).