Abstract
Durand de Gevigney and Szigeti (J Gr Theory 91(4):305–325, 2019) have recently given a min–max theorem for the (2, k)-connectivity augmentation problem. This article provides an \(O(n^3(m+ n \text { }\log \text { }n))\) time algorithm to find an optimal solution for this problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bang-Jensen, J., Frank, A., Jackson, B.: Preserving and increasing local edge-connectivity in mixed graphs. SIAM J. Discrete Math. 8(2), 155–178 (1995)
Bang-Jensen, J., Gabow, H., Jordán, T., Szigeti, Z.: Edge-connectivity augmentation with partition constraints. SIAM J. Discrete Math. 12(2), 160–207 (1999)
Bang-Jensen, J., Jackson, B.: Augmenting hypergraphs by edges of size two. Math. Program. 84(3), 467–481 (1999)
Bang-Jensen, J., Jordán, T.: Edge-connectivity augmentation preserving simplicity. SIAM J. Discrete Math. 11(4), 603–623 (1998)
Benczúr, A., Frank, A.: Covering symmetric supermodular functions by graphs. Math. Program. 84(3), 483–503 (1999)
Bernáth, A., Grappe, R., Szigeti, Z.: Augmenting the edge-connectivity of a hypergraph by adding a multipartite graph. J. Gr. Theory 72(3), 291–312 (2013)
Bernáth, A., Grappe, R., Szigeti, Z.: Partition constrained covering of a symmetric crossing supermodular function by a graph. SIAM J. Discrete Math. 31(1), 335–382 (2017)
Cai, G.R., Sun, Y.G.: The minimum augmentation of any graph to \(k\)-edge-connected graphs. Networks 19, 151–172 (1989)
Durand de Gevigney, O., Szigeti, Z.: On \((2, k)\)-connected graphs. J. Gr. Theory 91(4), 305–325 (2019)
Eswaran, K.P., Tarjan, R.E.: Augmentation problems. SIAM J. Comput. 5(4), 653–665 (1976)
Frank, A.: Augmenting graphs to meet edge-connectivity requirements. SIAM J. Discrete Math. 5(1), 22–53 (1992)
Frank, A.: Connections in Combinatorial Optimization. Oxford University Press, Oxford (2011)
Frank, A., Jordán, T.: Minimal edge-coverings of pairs of sets. J. Comb. Theory Ser. B 65(1), 73–110 (1995)
Jackson, B., Jordán, T.: Independence free graphs and vertex connectivity augmentation. J. Comb. Theory Ser. B 94(1), 31–77 (2005)
Kaneko, A., Ota, K.: On minimally \((n, \lambda )\)-connected graphs. J. Comb. Theory Ser. B 80(1), 156–171 (2000)
Nagamochi, H., Ibaraki, T.: Graph connectivity and its augmentation: applications of MA ordering. Discrete Appl. Math. 123, 447–472 (2002)
Szigeti, Z.: On edge-connectivity augmentation of graphs and hypergraphs. In: Cook, W., Lovász, L., Vygen, J. (eds.) Research Trends in Combinatorial Optimization, pp. 483–521. Springer, Berlin (2009)
Watanabe, T., Nakamura, A.: Edge-connectivity augmentation problems. Comput. Syst. Sci. 35, 96–144 (1987)
Acknowledgements
We thank the anonymous referees to kindly ask us to simplify our algorithm. This allowed us to find a new approach and to significantly simplify the algorithm and the proofs.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hörsch, F., Szigeti, Z. The (2, k)-Connectivity Augmentation Problem: Algorithmic Aspects. Algorithmica 83, 2333–2350 (2021). https://doi.org/10.1007/s00453-021-00829-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-021-00829-4