Advertisement

A polynomial-time algorithm for finding critical nodes in bipartite permutation graphs

  • Mohammed Lalou
  • Hamamache Kheddouci
Original Paper
  • 25 Downloads

Abstract

In this paper, we propose a polynomial-time algorithm for solving the Component-Cardinality-Constrained Critical Node Problem (3C-CNP) on bipartite permutation graphs. This problem, which is a variant of the well-known Critical Node Detection problem, consists in finding the minimal subset of nodes within a graph, the deletion of which results in a set of connected components of at most K nodes each one, where K is a given integer. The proposed algorithm is a dynamic programming scheme of time complexity \(O(nK^2)\), where n is the number of nodes. To provide evidences of algorithm’s efficiency, different experiments have been performed on randomly generated graphs.

Keywords

Critical nodes Graph connectivity Bipartite permutation graphs Dynamic programming Complete bipartite decomposition Graph separators 

Notes

References

  1. 1.
    Arulselvan, A., Commander, C.W., Shylo, O., Pardalos, P.M.: Cardinality-constrained critical node detection problem. In: Gülpınar, N., Harrison, P., Rüstem, B. (eds.) Performance Models and Risk Management in Communications Systems, pp. 79–91. Springer, New York (2011)CrossRefGoogle Scholar
  2. 2.
    Dinh, T.N., Xuan, Y., Thai, M.T., Park, E., Znati, T.: On approximation of new optimization methods for assessing network vulnerability. In: INFOCOM, 2010 Proceedings IEEE, pp. 1–9. IEEE (2010)Google Scholar
  3. 3.
    Kempe, D., Kleinberg, J., Tardos, É.: Influential nodes in a diffusion model for social networks. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) Automata, Languages and Programming. ICALP 2005. Lecture Notes in Computer Science, pp 1127–1138. Springer, Berlin (2005)Google Scholar
  4. 4.
    Lalou, M., Tahraoui, M.A., Kheddouci, H.: The critical node detection problem in networks: a survey. Comput. Sci. Rev. 28, 92–117 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Shen, S., Smith, J.C.: Polynomial-time algorithms for solving a class of critical node problems on trees and series-parallel graphs. Networks 60(2), 103–119 (2012)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Yannakakis, M.: Node-and edge-deletion NP-complete problems. In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, pp. 253–264. ACM (1978)Google Scholar
  7. 7.
    Addis, B., Di Summa, M., Grosso, A.: Identifying critical nodes in undirected graphs: complexity results and polynomial algorithms for the case of bounded treewidth. Discrete Appl. Math. 161(16), 2349–2360 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Arulselvan, A., Commander, C.W., Elefteriadou, L., Pardalos, P.M.: Detecting critical nodes in sparse graphs. Comput. Oper. Res. 36(7), 2193–2200 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Di Summa, M., Grosso, A., Locatelli, M.: Complexity of the critical node problem over trees. Comput. Oper. Res. 38(12), 1766–1774 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dinh, T.N., Thai, M.T.: Assessing attack vulnerability in networks with uncertainty. In: IEEE International Conference on Computer Communications (INFOCOM). IEEE (2015)Google Scholar
  11. 11.
    Aringhieri, R., Grosso, A., Hosteins, P., Scatamacchia, R.: A preliminary analysis of the distance based critical node problem. Electron. Notes Discrete Math. 55, 25–28 (2016)zbMATHCrossRefGoogle Scholar
  12. 12.
    Aringhieri, R., Grosso, A., Hosteins, P., Scatamacchia, R.: Polynomial and pseudo-polynomial time algorithms for different classes of the distance critical node problem. Discrete Appl. Math. (2018).  https://doi.org/10.1016/j.dam.2017.12.035
  13. 13.
    Veremyev, A., Prokopyev, O.A., Pasiliao, E.L.: Critical nodes for distance-based connectivity and related problems in graphs. Networks 66(3), 170–195 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lalou, M., Tahraoui, M.A., Kheddouci, H.: Component-cardinality-constrained critical node problem in graphs. Discrete Appl. Math. 210, 150–163 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Berger, A., Grigoriev, A., Zwaan, R.: Complexity and approximability of the k-way vertex cut. Networks 63(2), 170–178 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Shen, S., Smith, J.C., Goli, R.: Exact interdiction models and algorithms for disconnecting networks via node deletions. Discrete Optim. 9(3), 172–188 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Nguyen, D.T., Shen, Y., Thai, M.T.: Detecting critical nodes in interdependent power networks for vulnerability assessment. IEEE Trans. Smart Grid 4(1), 151–159 (2013)CrossRefGoogle Scholar
  18. 18.
    Aringhieri, R., Grosso, A., Hosteins, P., Scatamacchia, R.: A general evolutionary framework for different classes of critical node problems. Eng. Appl. Artif. Intell. 55, 128–145 (2016)zbMATHCrossRefGoogle Scholar
  19. 19.
    Dinh, T.N., Thai, M.T.: Precise structural vulnerability assessment via mathematical programming. In: Military Communications Conference, 2011-MILCOM 2011, pp. 1351–1356. IEEE (2011)Google Scholar
  20. 20.
    Dinh, T.N., Thai, M.T.: Network under joint node and link attacks: vulnerability assessment methods and analysis. IEEE/ACM Trans. Netw. 23(3), 1001–1011 (2015)CrossRefGoogle Scholar
  21. 21.
    Di Summa, M., Grosso, A., Locatelli, M.: Branch and cut algorithms for detecting critical nodes in undirected graphs. Comput. Optim. Appl. 53(3), 649–680 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Oosten, M., Rutten, J.H., Spieksma, F.C.: Disconnecting graphs by removing vertices: a polyhedral approach. Stat. Neerl. 61(1), 35–60 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Arulselvan, A., Commander, C.W., Pardalos, P.M., Shylo., O.: Managing network risk via critical node identification. In: Gulpinar, N., Rustem, B. (eds.) Risk Management in Telecommunication Networks. Springer, Berlin (2007)Google Scholar
  24. 24.
    Ventresca, M., Aleman, D.: A randomized algorithm with local search for containment of pandemic disease spread. Comput. Oper. Res. 48, 11–19 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Ventresca, M.: Global search algorithms using a combinatorial unranking-based problem representation for the critical node detection problem. Comput. Oper. Res. 39(11), 2763–2775 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Hermelin, D., Kaspi, M., Komusiewicz, C., Navon, B.: Parameterized complexity of critical node cuts. Theor. Comput. Sci. 651, 62–75 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Veremyev, A., Prokopyev, O.A., Pasiliao, E.L.: An integer programming framework for critical elements detection in graphs. J. Combin. Optim. 28(1), 233–273 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Brandes, U.: A faster algorithm for betweenness centrality. J. Math. Sociol. 25(2), 163–177 (2001)zbMATHCrossRefGoogle Scholar
  29. 29.
    Spinrad, J., Brandstädt, A., Stewart, L.: Bipartite permutation graphs. Discrete Appl. Math. 18(3), 279–292 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Uehara, R., Valiente, G.: Linear structure of bipartite permutation graphs and the longest path problem. Inf. Process. Lett. 2(103), 71–77 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Tomaino, V., Arulselvan, A., Veltri, P., Pardalos, P.M.: Studying connectivity properties in human protein-protein interaction network in cancer pathway. In: Pardalos, P., Xanthopoulos, P., Zervakis, M. (eds.) Data Mining for Biomarker Discovery, pp. 187–197. Springer, Boston (2012)CrossRefGoogle Scholar
  32. 32.
    Lalou, M., Kheddouci, H.: Least squares method for diffusion source localization in complex networks. In: Cherifi, H., Gaito, S., Quattrociocchi, W., Sala, A. (eds.) Complex Networks & Their Applications V, pp. 473–485. Springer, Cham (2016)Google Scholar
  33. 33.
    Lalou, M., Kheddouci, H., Hariri, S.: Identifying the cyber attack origin with partial observation: a linear regression based approach. In: 2017 IEEE 2nd International Workshops on Foundations and Applications of Self* Systems (FAS* W), pp. 329–333. IEEE (2017)Google Scholar
  34. 34.
    Michael, R.G., David, S.J.: Computers and Intractability: A Guide to the Theory of NP-Completeness. WH Freeman & Co., San Francisco (1979)zbMATHGoogle Scholar
  35. 35.
    Balas, E., de Souza, C.C.: The vertex separator problem: a polyhedral investigation. Math. Program. 103(3), 583–608 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Veremyev, A., Boginski, V., Pasiliao, E.L.: Exact identification of critical nodes in sparse networks via new compact formulations. Optim. Lett. 8(4), 1245–1259 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Pullan, W.: Heuristic identification of critical nodes in sparse real-world graphs. J. Heuristics 21(5), 577–598 (2015)CrossRefGoogle Scholar
  38. 38.
    Saitoh, T., Otachi, Y., Yamanaka, K., Uehara, R.: Random generation and enumeration of bipartite permutation graphs. J. Discrete Algorithms 10, 84–97 (2012)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université Claude Bernard Lyon 1, France - UFR-InformatiqueVilleurbanne CedexFrance
  2. 2.Institut des Sciences et TechnologieCentre Universitaire A. BoussoufMilaAlgeria
  3. 3.Département d’Informatique, Faculté des Sciences ExactesUniversité de BéjaiaBejaiaAlgeria

Personalised recommendations