Approximating connectivity augmentation problems

Abstract

Let G = (V, E) be an undirected graph and let S ⊆ V . The S-connectivity λ S 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 λ S 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 = max u,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(2 log 1−ε n ) for any fixed ε > 0, unless NP ⊆ DTIME(n polylog(n) ).

Augmenting Edge-Connectivity between Vertex Subsets

“…As for LAP, Frank (1992Frank ( , 1994 showed that it ). * * 7/4-approximable if r depends only on W 2 (Nutov 2005), while solvable in O(n 3 |W 2 |(m + n log n)) time if in addition r ≥ 2 holds (Ishii and Hagiwara 2006). …”

Inapproximability of Survivable Networks

“…It seems that this result of [13] cannot be deduced from our work, as the proof of the directed case is essentially the same as that of the undirected one. We also note that the hardness result in Corollary 1.3 was recently improved by Chakraborty, Chuzhoy, and Khanna [1] to a k ε -hardness of approximation for all k > k 0 , where ε, k 0 are universal constants.…”

“…In [13] it is proved that the same hardness result holds even for 0, 1-costs, for both directed and undirected graphs (for large values of k). It seems that this result of [13] cannot be deduced from our work, as the proof of the directed case is essentially the same as that of the undirected one.…”

“…Specifically, Dodis and Khanna [4] showed that Directed Steiner Forest is at least as hard to approximate as Label-Cover. By extending the construction of [4], Kortsarz, Krauthgamer, and Lee [9] showed a similar hardness result for Undirected {0, k}-SNDP; the same hardness is valid even for 0, 1-costs, see [13]. Even a very special case of Directed Steiner Tree -the Group Steiner Tree problem on trees, is unlikely to admit a log 2−ε n ratio for any ε > 0 [7].…”

“…• Directed Steiner Forest with 0, 1-costs admits an O(log n)-approximation [11], while the undirected variant is unlikely to admit a polylogarithmic approximation [13].…”