Skip to main content
SpringerLink
Log in

On the Fixed-Parameter Tractability of the Maximum Connectivity Improvement Problem

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

In the Maximum Connectivity Improvement (MCI) problem, we are given a directed graph G = (V,E) and an integer B and we are asked to find B new edges to be added to G in order to maximize the number of connected pairs of vertices in the resulting graph. The MCI problem has been studied from the approximation point of view. In this paper, we approach it from the parameterized complexity perspective in the case of directed acyclic graphs. We show several hardness and algorithmic results with respect to different natural parameters. Our main result is that the problem is W[2]-hard for parameter B and it is FPT for parameters |V |− B and ν, the matching number of G. We further characterize the MCI problem with respect to other complementary parameters.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. One can assume that |V | is an upper bound for B as otherwise the graph can be made strongly connected (see Theorem 1).

  2. We can equivalently define MCI as a decision problem without affecting (up to a poly-logarithmic factor) the complexity of the algorithms given in this paper.

References

  1. Avrachenkov, K., Litvak, N.: The effect of new links on google pagerank. Stoc. Models 22(2), 319–331 (2006)

    Article  MathSciNet  Google Scholar 

  2. Bang-Jensen, J., Basavaraju, M., Klinkby, K.V., Misra, P., Ramanujan, M.S., Saurabh, S., Zehavi, M.: Parameterized algorithms for survivable network design with uniform demands. In: 29Th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 2838–2850 (2018)

  3. Bang-Jensen, J., Gutin, G.Z.: Digraphs: Theory, Algorithms and Applications. Springer (2008)

  4. Basavaraju, M., Fomin, F.V., Golovach, P., Misra, P., Ramanujan, M.S., Saurabh, S.: Parameterized algorithms to preserve connectivity. In: 41St International Colloquium on Automata, Languages and Programming (ICALP), pp. 800–811 (2014)

  5. Bergamini, E., Crescenzi, P., D’Angelo, G., Meyerhenke, H., Severini, L., Velaj, Y.: Improving the betweenness centrality of a node by adding links. ACM Journal of Experimental Algorithmics 23 (2018)

  6. Cesati, M.: Compendium of parameterized problems. Department of Computer Science, Systems, and Industrial Engineering, University of Rome Tor Vergata 22 (2006)

  7. Cheriyan, J., Végh, L.: Approximating minimum-cost k-node connected subgraphs via independence-free graphs. SIAM J. Comput. 43(4), 1342–1362 (2014)

    Article  MathSciNet  Google Scholar 

  8. Corò, F., D’Angelo, G., Pinotti, C.M.: On the maximum connectivity improvement problem. In: 14Th International Symposium on Algorithms and Experiments for Wireless Networks (ALGOSENSOR) (2018)

  9. Crescenzi, P., D’Angelo, G., Severini, L., Velaj, Y.: Greedily improving our own centrality in a network. In: Proceedings of the 14th International Symposium on Experimental Algorithms (SEA 2015), LNCS, vol. 9125, pp. 43–55. Springer (2015)

  10. Crescenzi, P., D’Angelo, G., Severini, L., Velaj, Y.: Greedily improving our own closeness centrality in a network. TKDD 11(1), 9:1–9:32 (2016)

    Article  Google Scholar 

  11. Cygan, M., Kortsarz, G., Nutov, Z.: Steiner forest orientation problems. SIAM J. Discrete Math. 27(3), 1503–1513 (2013)

    Article  MathSciNet  Google Scholar 

  12. D’Angelo, G., Olsen, M., Severini, L.: Coverage centrality maximization in undirected networks. In: 33rd AAAI Conference on Artificial Intelligence (AAAI-19). To appear. CoRR arXiv:1811.04331 (2019)

  13. Demaine, E.D., Zadimoghaddam, M.: Minimizing the diameter of a network using shortcut edges. In: 12Th Scandinavian Symp. and Work. on Algorithm Theory (SWAT), LNCS, vol. 6139, pp. 420–431. Springer (2010)

  14. Eswaran, K.P., Tarjan, R.E.: Augmentation problems. SIAM J. Comput. 5(4), 653–665 (1976)

    Article  MathSciNet  Google Scholar 

  15. Frank, A.: Connections in Combinatorial Optimization. Oxford University Press (2011)

  16. Gao, Y., Hare, D.R., Nastos, J.: The parametric complexity of graph diameter augmentation. Discret. Appl. Math. 161(10-11), 1626–1631 (2013)

    Article  MathSciNet  Google Scholar 

  17. Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)

    MATH  Google Scholar 

  18. Gutin, G., Ramanujan, M., Reidl, F., Wahlström, M.: Path-contractions, edge deletions and connectivity preservation. J. Comput. Syst. Sci. 101, 1–20 (2019)

    Article  MathSciNet  Google Scholar 

  19. Hoffmann, C., Molter, H., Sorge, M.: The parameterized complexity of centrality improvement in networks. In: 44Th International Conference on Current Trends in Theory and Practice of Computer Science(SOFSEM 2018), pp. 111–124 (2018)

  20. Kortsarz, G., Nutov, Z.: Approximating minimum-cost connectivity problems. In: Gonzalez, T.F. (ed.) Handbook of Approximation Algorithms and Metaheuristics. Chapman and Hall/CRC (2007)

  21. Marx, D., Végh, L.A.: Fixed-parameter algorithms for minimum-cost edge-connectivity augmentation. ACM Trans. Algorithms 11(4), 27:1–27:24 (2015)

    Article  MathSciNet  Google Scholar 

  22. Meyerson, A., Tagiku, B.: Minimizing average shortest path distances via shortcut edge addition. In: 13Th Int. Work. on Approx. Alg. for Comb. Opt. Prob. (APPROX), LNCS, vol. 5687, pp. 272–285. Springer (2009)

  23. Nutov, Z.: Improved approximation algorithms for minimum cost node-connectivity augmentation problems. Theor. Comput. Syst. 62(3), 510–532 (2018)

    Article  MathSciNet  Google Scholar 

  24. Nutov, Z., Gonzalez, T.F.: Node-connectivity survivable network problems. In: Handbook of Approximation Algorithms and Metaheuristics Contemporary and Emerging Applications, vol. 2. Taylor and Francis Group (2018)

  25. Olsen, M., Viglas, A.: On the approximability of the link building problem. Theor. Comput. Sci. 518, 96–116 (2014)

    Article  MathSciNet  Google Scholar 

  26. Papagelis, M.: Refining social graph connectivity via shortcut edge addition. ACM Trans. Knowl. Discov. Data (TKDD) 10(2), 12 (2015)

    Google Scholar 

  27. Pilipczuk, M., Wahlström, M.: Directed multicut is w[1]-hard, even for four terminal pairs. ACM Trans. Comput. Theory 10(3), 13:1–13:18 (2018)

    Article  MathSciNet  Google Scholar 

  28. Raghavan, S.: A note on eswaran and tarjan’s algorithm for the strong connectivity augmentation. The Next Wave in Computing, Optimization, and Decision Technologies, pp. 19–26 (2005)

  29. Sasák, R.: Comparing 17 Graph Parameters. Master thesis, University of Bergen (2010)

Download references

Author information

Authors and Affiliations

  1. Sapienza University of Rome, Rome, Italy

    Federico Corò

  2. Gran Sasso Science Institute, L’Aquila, Italy

    Gianlorenzo D’Angelo & Vahan Mkrtchyan

Authors
  1. Federico Corò
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gianlorenzo D’Angelo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vahan Mkrtchyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Federico Corò.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work has been partially supported by the Italian MIUR PRIN 2017 Project ALGADIMAR “Algorithms, Games, and Digital Markets.”

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Corò, F., D’Angelo, G. & Mkrtchyan, V. On the Fixed-Parameter Tractability of the Maximum Connectivity Improvement Problem. Theory Comput Syst 64, 1094–1109 (2020). https://doi.org/10.1007/s00224-020-09977-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-020-09977-6

Keywords

Search

Navigation