Abstract
The notion of element-connectivity has found several important applications in network design and routing problems. We focus on a reduction step that preserves the element-connectivity [18, 4, 3], which when applied repeatedly allows one to reduce the original graph to a simpler one. This pre-processing step is a crucial ingredient in several applications. In this paper we revisit this reduction step and provide a new proof via the use of setpairs. Our main contribution is algorithmic results for several basic problems on element-connectivity including the problem of achieving the aforementioned graph simplification. We utilize the underlying submodularity properties of element-connectivity to derive faster algorithms.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Aazami, A., Cheriyan, J., Jampani, K.: Approximation Algorithms and Hardness Results for Packing Element-Disjoint Steiner Trees in Planar Graphs. Algorithmica 63(1–2), 425–456 (2012)
Benczúr, A.A.: Counterexamples for directed and node capacitated cut-trees. SIAM Journal on Computing 24(3), 505–510 (1995)
Chekuri, C., Korula, N.: A graph reduction step preserving element-connectivity and packing steiner trees and forests. SIAM Journal on Discrete Mathematics 28(2), 577–597 (2014)
Cheriyan, J., Salavatipour, M.: Packing Element-Disjoint Steiner Trees. ACM Transactions on Algorithms 3(4) (2007)
Cheriyan, J., Vempala, S., Vetta, A.: Network Design via Iterative Rounding of Setpair Relaxations. Combinatorica 26(3), 255–275 (2006)
Cheung, H.Y., Lau, L.C., Leung, K.M.: Graph connectivities, network coding, and expander graphs. In: Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, pp. 190–199 (2011)
Duan, R.: Breaking the O(n 2.5) Deterministic Time Barrier for Undirected Unit-Capacity Maximum Flow. In: Proceedings of ACM-SIAM SODA, pp. 1171–1179 (2013)
Even, S., Tarjan, R.: Network flow and testing graph connectivity. SIAM Journal on Computing 4(4), 507–518 (1975)
Fleischer, L., Jain, K., Williamson, D.: Iterative Rounding 2-Approximation Algorithms for Minimum-Cost Vertex Connectivity Problems. Journal of Computer and System Sciences 72(5), 838–867 (2006)
Frank, A., Ibaraki, T., Nagamochi, H.: On sparse subgraphs preserving connectivity properties. Journal of Graph Theory 17(3), 275–281 (1993)
Gabow, H.N.: Efficient splitting off algorithms for graphs. In: Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, STOC 1994, pp. 696–705. ACM, New York (1994)
Gabow, H.N.: Using expander graphs to find vertex connectivity. In: Proc. 41st Annual IEEE Symposium on Foundations of Computer Science, pp. 410–420 (2000)
Gabow, H.N., Tarjan, R.E.: Algorithms for Two Bottleneck Optimization Problems. Journal of Algorithms 9(3), 411–417 (1988)
Gabow, H., Sankowski, P.: Algebraic algorithms for b-matching, shortest undirected paths, and f-factors. In: 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS), pp. 137–146 (October 2013)
Hao, J., Orlin, J.B.: A faster algorithm for finding the minimum cut in a graph. In: Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1992, pp. 165–174. Society for Industrial and Applied Mathematics, Philadelphia (1992)
Hariharan, R., Kavitha, T., Panigrahi, D., Bhalgat, A.: An Õ(mn) Gomory-Hu tree construction algorithm for unweighted graphs. In: Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, STOC 2007, pp. 605–614. ACM (2007)
Henzinger, M.R., Rao, S., Gabow, H.N.: Computing vertex connectivity: New bounds from old techniques. Journal of Algorithms 34(2), 222–250 (2000)
Hind, H., Oellermann, O.: Menger-Type Results for Three or More Vertices. Congressus Numerantium, pp. 179–204 (1996)
Jain, K., Mǎndoiu, I., Vazirani, V., Williamson, D.: A Primal-Dual Schema Based Approximation Algorithm for the Element Connectivity Problem. In: Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 484–489 (1999)
Karger, D.: Random Sampling in Graph Optimization Problems. Ph.D. thesis, Stanford University (1994)
Kawarabayashi, K., Thorup, M.: Deterministic Global Minimum Cut of a Simple Graph in Near-Linear Time. In: Proceedings of ACM STOC, pp. 665–674 (2015)
Klimmek, R., Wagner, F.: A simple hypergraph min cut algorithm. Tech. Rep. B 96-02, Bericht FU Berlin Fachbereich Mathematik und Informatik (1996)
Lau, L.C., Yung, C.K.: Efficient edge splitting-off algorithms maintaining all-pairs edge-connectivities. SIAM Journal on Computing 42(3), 1185–1200 (2013)
Lovász, L.: On Some Connectivity Properties of Eulerian Graphs. Acta Mathematica Hungarica 28(1), 129–138 (1976)
Mader, W.: A Reduction Method for Edge-Connectivity in Graphs. Annals of Discrete Mathematics 3, 145–164 (1978)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chekuri, C., Rukkanchanunt, T., Xu, C. (2015). On Element-Connectivity Preserving Graph Simplification. In: Bansal, N., Finocchi, I. (eds) Algorithms - ESA 2015. Lecture Notes in Computer Science(), vol 9294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48350-3_27
Download citation
DOI: https://doi.org/10.1007/978-3-662-48350-3_27
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-48349-7
Online ISBN: 978-3-662-48350-3
eBook Packages: Computer ScienceComputer Science (R0)