Fast Exact Algorithms for Survivable Network Design with Uniform Requirements

  • Akanksha Agrawal
  • Pranabendu Misra
  • Fahad Panolan
  • Saket Saurabh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

Abstract

We design exact algorithms for the following two problems in survivable network design: (i) designing a minimum cost network with a desired value of edge connectivity, which is called Minimum Weight\(\lambda \)-connected Spanning Subgraph and (ii) augmenting a given network to a desired value of edge connectivity at a minimum cost which is called Minimum Weight\(\lambda \)-connectivity Augmentation. Many well known problems such as Minimum Spanning Tree, Hamiltonian Cycle, Minimum 2-Edge Connected Spanning Subgraph and Minimum Equivalent Digraph reduce to these problems in polynomial time. It is easy to see that a minimum solution to these problems contains at most \(2 \lambda (n-1)\) edges. Using this fact one can design a brute-force algorithm which runs in time \(2^{\mathcal {O}(\lambda n(\log n + \log \lambda )}\). However no better algorithms were known. In this paper, we give the first single exponential time algorithm for these problems, i.e. running in time \(2^{\mathcal {O}(\lambda n)}\), for both undirected and directed networks. Our results are obtained via well known characterizations of \(\lambda \)-connected graphs, their connections to linear matroids and the recently developed technique of dynamic programming with representative sets.

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References

  1. 1.
    Bang-Jensen, J., Gutin, G.Z.: Digraphs: theory, algorithms and applications. Springer Science & Business Media (2008)Google Scholar
  2. 2.
    Basavaraju, M., Fomin, F.V., Golovach, P., Misra, P., Ramanujan, M.S., Saurabh, S.: Parameterized algorithms to preserve connectivity. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 800–811. Springer, Heidelberg (2014). doi:10.1007/978-3-662-43948-7_66 Google Scholar
  3. 3.
    Bellman, R.: Dynamic programming treatment of the travelling salesman problem. Journal of the ACM (JACM) 9(1), 61–63 (1962)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Berman, P., DasGupta, B., Karpinski, M.: Approximating transitive reductions for directed networks. In: Dehne, F., Gavrilova, M., Sack, J.R., Tóth, C.D. (eds.) WADS 2009. LNCS, vol. 5664, pp. 74–85. Springer, Heidelberg (2009). doi:10.1007/978-3-642-03367-4_7
  5. 5.
    Bjorklund, A.: Determinant sums for undirected hamiltonicity. SIAM Journal on Computing 43(1), 280–299 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cheriyan, J., Thurimella, R.: Approximating minimum-size k-connected spanning subgraphs via matching. SIAM Journal on Computing 30(2), 528–560 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cygan, M., Fomin, F.V., Golovnev, A., Kulikov, A.S., Mihajlin, I., Pachocki, J., Socala, A.: Tight bounds for graph homomorphism and subgraph isomorphism. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10–12, pp. 1643–1649 (2016)Google Scholar
  8. 8.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer (2015)Google Scholar
  9. 9.
    Cygan, M., Kratsch, S., Nederlof, J.: Fast hamiltonicity checking via bases of perfect matchings. In: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, pp. 301–310. ACM (2013)Google Scholar
  10. 10.
    Fomin, F.V., Lokshtanov, D., Panolan, F., Saurabh, S.: Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM 63(4), 29: 1–29: 60 (2016). http://doi.acm.org/10.1145/2886094
  11. 11.
    Frank, A.: Augmenting graphs to meet edge-connectivity requirements. SIAM Journal on Discrete Mathematics 5(1), 25–53 (1992)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Frank, H., Chou, W.: Connectivity considerations in the design of survivable networks. IEEE Transactions on Circuit Theory 17(4), 486–490 (1970)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gabow, H.N.: A matroid approach to finding edge connectivity and packing arborescences. Journal of Computer and System Sciences 50(2), 259–273 (1995)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gusfield, D.: A graph theoretic approach to statistical data security. SIAM Journal on Computing 17(3), 552–571 (1988)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. Journal of the Society for Industrial and Applied Mathematics 10(1), 196–210 (1962)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Jain, S., Gopal, K.: On network augmentation. Reliability, IEEE Transactions on 35(5), 541–543 (1986)CrossRefMATHGoogle Scholar
  17. 17.
    Kao, M.Y.: Data security equals graph connectivity. SIAM Journal on Discrete Mathematics 9(1), 87–100 (1996)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Khuller, S.: Approximation algorithms for finding highly connected subgraphs. Vertex 2, 2 (1997)Google Scholar
  19. 19.
    Khuller, S., Vishkin, U.: Biconnectivity approximations and graph carvings. Journal of the ACM (JACM) 41(2), 214–235 (1994)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kortsarz, G., Nutov, Z.: Approximating minimum cost connectivity problems. In: Dagstuhl Seminar Proceedings. Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2010)Google Scholar
  21. 21.
    Marx, D.: A parameterized view on matroid optimization problems. Theor. Comput. Sci. 410(44), 4471–4479 (2009). http://dx.doi.org/10.1016/j.tcs.2009.07.027
  22. 22.
    Marx, D., Végh, L.A.: Fixed-parameter algorithms for minimum-cost edge-connectivity augmentation. ACM Transactions on Algorithms (TALG) 11(4), 27 (2015)MathSciNetMATHGoogle Scholar
  23. 23.
    Moyles, D.M., Thompson, G.L.: An algorithm for finding a minimum equivalent graph of a digraph. Journal of the ACM (JACM) 16(3), 455–460 (1969)CrossRefMATHGoogle Scholar
  24. 24.
    Schrijver, A.: Combinatorial optimization: polyhedra and efficiency, vol. 24. Springer Science & Business Media (2003)Google Scholar
  25. 25.
    Watanabe, T., Narita, T., Nakamura, A.: 3-edge-connectivity augmentation problems. In: IEEE International Symposium on Circuits and Systems, pp. 335–338. IEEE (1989)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Akanksha Agrawal
    • 1
  • Pranabendu Misra
    • 1
  • Fahad Panolan
    • 1
  • Saket Saurabh
    • 1
    • 2
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.The Institute of Mathematical SciencesHBNIChennaiIndia

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