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.
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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)
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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
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DOI: https://doi.org/10.1007/978-3-662-48350-3_27
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