Abstract
We consider the (undirected) Node Connectivity Augmentation (NCA) problem: given a graph J = (V,E J ) and connectivity requirements {r(u,v):u,v ∈ V}, find a minimum size set I of new edges (any edge is allowed) so that J + I contains r(u,v) internally disjoint uv-paths, for all u,v ∈ V. In the Rooted NCA there is s ∈ V so that r(u,v) > 0 implies u = s or v = s. For large values of k = max u,v ∈ V r(u,v), NCA is at least as hard to approximate as Label-Cover and thus it is unlikely to admit a polylogarithmic approximation. Rooted NCA is at least as hard to approximate as Hitting-Set. The previously best approximation ratios for the problem were O(k ln n) for NCA and O(ln n) for Rooted NCA. In [Approximating connectivity augmentation problems, SODA 2005] the author posed the following open question: Does there exist a function ρ(k) so that NCA admits a ρ(k)-approximation algorithm? In this paper we answer this question, by giving an approximation algorithm with ratios O(k ln 2 k) for NCA and O(ln 2 k) for Rooted NCA. This is the first approximation algorithm with ratio independent of n, and thus is a constant for any fixed k. Our algorithm is based on the following new structural result which is of independent interest. If \({\cal D}\) is a set of node pairs in a graph J, then the maximum degree in the hypergraph formed by the inclusion minimal tight sets separating at least one pair in \({\cal D}\) is O(ℓ2), where ℓ is the maximum connectivity of a pair in \({\cal D}\).
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Nutov, Z. (2009). Approximating Node-Connectivity Augmentation Problems. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_22
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DOI: https://doi.org/10.1007/978-3-642-03685-9_22
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