Abstract
A graph is called ℓ-connected from U to r if there are ℓ internally disjoint paths from every node u ∈ U to r. The Rooted Subset Connectivity Augmentation Problem (RSCAP) is as follows: given a graph G=(V+r,E), a node subset U ⊆ V, and an integer k, find a smallest set F of new edges such that G+F is k-connected from U to r. In this paper we consider mainly a restricted version of RSCAP in which the input graph G is already (k-1)-connected from U to r. For this version we give an O(ln |U|)-approximation algorithm, and show that the problem cannot achieve a better approximation guarantee than the Set Cover Problem (SCP) on |U| elements and with |V|-|U| sets. For the general version of RSCAP we give an O(ln k ln |U|)-approximation algorithm. For U=V we get the Rooted Connectivity Augmentation Problem (RCAP). For directed graphs RCAP is polynomially solvable, but for undirected graphs its complexity status is not known: no polynomial algorithm is known, and it is also not known to be NP-hard. For undirected graphs with the input graph G being (k-1)-connected from V to r, we give an algorithm that computes a solution of size exceeding a lower bound of the optimum by at most (k-1)/2 edges.
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© 2003 Springer-Verlag Berlin Heidelberg
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Nutov, Z. (2003). Approximating Rooted Connectivity Augmentation Problems. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds) Approximation, Randomization, and Combinatorial Optimization.. Algorithms and Techniques. RANDOM APPROX 2003 2003. Lecture Notes in Computer Science, vol 2764. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45198-3_13
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DOI: https://doi.org/10.1007/978-3-540-45198-3_13
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