Your Rugby Mates Don’t Need to Know Your Colleagues: Triadic Closure with Edge Colors

  • Laurent Bulteau
  • Niels GrüttemeierEmail author
  • Christian Komusiewicz
  • Manuel Sorge
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11485)


Given an undirected graph \(G=(V,E)\) the NP-hard Strong Triadic Closure (STC) problem asks for a labeling of the edges as weak and strong such that at most k edges are weak and for each induced \(P_3\) in G at least one edge is weak. In this work, we study the following generalizations of STC with c different strong edge colors. In Multi-STC an induced \(P_3\) may receive two strong labels as long as they are different. In Edge-List Multi-STC and Vertex-List Multi-STC we may additionally restrict the set of permitted colors for each edge of G. We show that, under the ETH, Edge-List Multi-STC and Vertex-List Multi-STC cannot be solved in time \(2^{o(|V|^2)}\), and that Multi-STC is NP-hard for every fixed c. We then extend previous fixed-parameter tractability results and kernelizations for STC to the three variants with multiple edge colors or outline the limits of such an extension.


  1. 1.
    Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theor. Comput. Sci. 411(40–42), 3736–3756 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). Scholar
  3. 3.
    Diehl, C.P., Namata, G., Getoor, L.: Relationship identification for social network discovery. In: Proceedings of the 22nd AAAI, pp. 546–552. AAAI Press (2007)Google Scholar
  4. 4.
    Gallai, T.: Transitiv orientierbare Graphen. Acta Math. Hung. 18(1–2), 25–66 (1967)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Golovach, P.A., Heggernes, P., Konstantinidis, A.L., Lima, P.T., Papadopoulos, C.: Parameterized aspects of strong subgraph closure. In: Proceedings of the 16th SWAT. LIPIcs, vol. 101, pp. 23:1–23:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2018)Google Scholar
  6. 6.
    Granovetter, M.: The strength of weak ties. Am. J. Sociol. 78, 1360–1380 (1973)CrossRefGoogle Scholar
  7. 7.
    Grüttemeier, N., Komusiewicz, C.: On the relation of strong triadic closure and cluster deletion. In: Brandstädt, A., Köhler, E., Meer, K. (eds.) WG 2018. LNCS, vol. 11159, pp. 239–251. Springer, Cham (2018). Scholar
  8. 8.
    Hsu, W.L., Ma, T.H.: Substitution decomposition on chordal graphs and applications. In: Hsu, W.L., Lee, R.C.T. (eds.) ISA 1991. LNCS, vol. 557, pp. 52–60. Springer, Heidelberg (1991). Scholar
  9. 9.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kierstead, H.A., Kostochka, A.V., Mydlarz, M., Szemerédi, E.: A fast algorithm for equitable coloring. Combinatorica 30(2), 217–224 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Konstantinidis, A.L., Nikolopoulos, S.D., Papadopoulos, C.: Strong triadic closure in cographs and graphs of low maximum degree. Theor. Comput. Sci. 740, 76–84 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kowalik, L., Lauri, J., Socala, A.: On the fine-grained complexity of rainbow coloring. SIAM J. Discrete Math. 32, 1672–1705 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kowalik, L., Socala, A.: Tight lower bounds for list edge coloring. In: Proceedings of the 16th SWAT. LIPIcs, vol. 101, pp. 28:1–28:12. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2018)Google Scholar
  14. 14.
    Protti, F., da Silva, M.D., Szwarcfiter, J.L.: Applying modular decomposition to parameterized cluster editing problems. Theory Comput. Syst. 44(1), 91–104 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Rozenshtein, P., Tatti, N., Gionis, A.: Inferring the strength of social ties: a community-driven approach. In: Proceedings of the 23rd KDD, pp. 1017–1025. ACM (2017)Google Scholar
  16. 16.
    Sintos, S., Tsaparas, P.: Using strong triadic closure to characterize ties in social networks. In: Proceedings of the 20th KDD, pp. 1466–1475. ACM (2014)Google Scholar
  17. 17.
    Tang, J., Lou, T., Kleinberg, J.M.: Inferring social ties across heterogenous networks. In: Proceedings of the 5th WSDM, pp. 743–752. ACM (2012)Google Scholar
  18. 18.
    Tovey, C.A.: A simplified NP-complete satisfiability problem. Discrete Appl. Math. 8(1), 85–89 (1984)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Laurent Bulteau
    • 1
  • Niels Grüttemeier
    • 2
    Email author
  • Christian Komusiewicz
    • 2
  • Manuel Sorge
    • 3
  1. 1.CNRS, Université Paris-Est Marne-la-ValléeParisFrance
  2. 2.Fachbereich Mathematik und InformatikPhilipps-Universität MarburgMarburgGermany
  3. 3.Faculty of Mathematics, Informatics and MechanicsUniversity of WarsawWarsawPoland

Personalised recommendations