Approximating connectivity augmentation problems

Let <i>G</i> = (<i>V, E</i>) be a graph and let <i>S</i> ⊆ <i>V.</i> The <i>S-connectivity</i> λ<i>s</i>(<i>u, v; G</i>) of <i>u</i> and <i>v</i> in <i>G</i> is the maximum number of <i>uv</i>-paths that no two of them have an edge or a node in <i>S</i> - {<i>u, v</i>} in common. The corresponding <i>Connectivity Augmentation Problem (CAP)</i> is: given a graph <i>G</i> = (<i>V, E</i>), a node subset <i>S</i> ⊆ <i>V</i>, and a nonnegative integer requirement function <i>r</i>(<i>u, v</i>) on the set of pairs of nodes, add a minimum size set <i>F</i> of new edges to <i>G</i> so that λ<i>s</i>(<i>u, v; G</i> + <i>F</i>) ≥ <i>r</i>(<i>u, v</i>) holds for all <i>u, v</i> ∈ <i>V</i>. Three extensively studied particular cases are: the <i>edge-</i> (<i>S</i> = θ), the <i>node-</i> (<i>S = V</i>), and the <i>element-</i> (<i>r</i>(<i>u, v</i>) = 0 whenever <i>u</i> ∈ <i>S</i> or <i>v</i> ∈ <i>S</i>) CAP. A polynomial algorithm for edge- CAP was developed by A. Frank [8]. In this paper we consider the element-CAP and the node-CAP, that are NP- hard even for <i>r</i>(<i>u, v</i>) ∈ {0, 2}. Our main result is a 7/4- approximation algorithm for the element-CAP, improving the previously best known 2-approximation. For the {0, <i>k</i>}- element-CAP (with <i>r</i>(<i>u, v</i>) ∈ {0, <i>k</i>}) and for the {0, 1, 2}-element-CAP we give a 3/2-approximation algorithm. The approximation ratios are based on a new lower bound on the number of edges needed to cover a skew-supermodular set function. For the node-CAP we establish the following approximation threshold: the {0, <i>k</i>}-node-CAP cannot be approximated within <i>O</i>(2^{log-1-∈<i>n</i>}) for any fixed ∈ > 0, unless NP ⊆ DTIME(<i>n</i>^{Polylog(<i>n</i>)}); thus the node-CAP is unlikely to have a polylogarithmic approximation.