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-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 ( r max ṡ ln n ) for Node-CA, where r max = 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 ) ).