Parameterized Aspects of Triangle Enumeration

  • Matthias BentertEmail author
  • Till Fluschnik
  • André Nichterlein
  • Rolf Niedermeier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10472)


Listing all triangles in an undirected graph is a fundamental graph primitive with numerous applications. It is trivially solvable in time cubic in the number of vertices. It has seen a significant body of work contributing to both theoretical aspects (e.g., lower and upper bounds on running time, adaption to new computational models) as well as practical aspects (e.g. algorithms tuned for large graphs). Motivated by the fact that the worst-case running time is cubic, we perform a systematic parameterized complexity study of triangle enumeration, providing both positive results (new enumerative kernelizations, “subcubic” parameterized solving algorithms) as well as negative results (uselessness in terms of possibility of “faster” parameterized algorithms of certain parameters such as diameter).


  1. 1.
    Abboud, A., Williams, V.V.: Popular conjectures imply strong lower bounds for dynamic problems. In: Proceedings of the 55th FOCS, pp. 434–443. IEEE Computer Society (2014)Google Scholar
  2. 2.
    Abboud, A., Williams, V.V., Wang, J.R.: Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In: Proceedings of the 27th SODA, pp. 377–391. SIAM (2016)Google Scholar
  3. 3.
    Bar-Yehuda, R., Geiger, D., Naor, J., Roth, R.M.: Approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and Bayesian inference. SIAM J. Comput. 27(4), 942–959 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Becchetti, L., Boldi, P., Castillo, C., Gionis, A.: Efficient semi-streaming algorithms for local triangle counting in massive graphs. In: Proceedings of the 14th ACM KDD, pp. 16–24. ACM (2008)Google Scholar
  5. 5.
    Björklund, A., Pagh, R., Williams, V.V., Zwick, U.: Listing triangles. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 223–234. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-43948-7_19 Google Scholar
  6. 6.
    Bretscher, A., Corneil, D.G., Habib, M., Paul, C.: A simple linear time LexBFS cograph recognition algorithm. SIAM J. Discret. Math. 22(4), 1277–1296 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chiba, N., Nishizeki, T.: Arboricity and subgraph listing algorithms. SIAM J. Comput. 14(1), 210–223 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM J. Comput. 14(4), 926–934 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Creignou, N., Meier, A., Müller, J.S., Schmidt, J., Vollmer, H.: Paradigms for parameterized enumeration. Theory Comput. Syst. 60(4), 737–758 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Doucha, M., Kratochvíl, J.: Cluster vertex deletion: a parameterization between vertex cover and clique-width. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 348–359. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-32589-2_32 CrossRefGoogle Scholar
  11. 11.
    Ferrara, E.: Measurement and analysis of online social networks systems. In: Alhajj, R., Rokne, J. (eds.) Encyclopedia of Social Network Analysis and Mining, pp. 891–893. Springer, New York (2014). doi: 10.1007/978-1-4614-6170-8_242 Google Scholar
  12. 12.
    Fluschnik, T., Komusiewicz, C., Mertzios, G.B., Nichterlein, A., Niedermeier, R., Talmon, N.: When can graph hyperbolicity be computed in linear time? In: Ellen F., Kolokolova A., Sack J.R. (eds.) Proceedings of the 15th WADS. LNCS, vol. 10389, pp. 397–408. Springer, Heidelberg (2017). doi: 10.1007/978-3-319-62127-2_34. ISBN 978-3-319-62126-5
  13. 13.
    Fomin, F.V., Lokshtanov, D., Pilipczuk, M., Saurabh, S., Wrochna, M.: Fully polynomial-time parameterized computations for graphs and matrices of low treewidth. In: Proceedings of the 28th SODA, pp. 1419–1432. SIAM (2017)Google Scholar
  14. 14.
    Giannopoulou, A.C., Mertzios, G.B., Niedermeier, R.: Polynomial fixed-parameter algorithms: a case study for longest path on interval graphs. In: Proceedings of the 10th IPEC, LIPIcs, vol. 43, pp. 102–113. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015)Google Scholar
  15. 15.
    Grabow, C., Grosskinsky, S., Kurths, J., Timme, M.: Collective relaxation dynamics of small-world networks. Phys. Rev. E 91, 052815 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Green, O., Bader, D.A.: Faster clustering coefficient using vertex covers. In: Proceedings of the 6th SocialCom, pp. 321–330. IEEE Computer Society (2013)Google Scholar
  17. 17.
    Habib, M., Paul, C., Viennoti, L.: A synthesis on partition refinement: a useful routine for strings, graphs, boolean matrices and automata. In: Morvan, M., Meinel, C., Krob, D. (eds.) STACS 1998. LNCS, vol. 1373, pp. 25–38. Springer, Heidelberg (1998). doi: 10.1007/BFb0028546 Google Scholar
  18. 18.
    Itai, A., Rodeh, M.: Finding a minimum circuit in a graph. SIAM J. Comput. 7(4), 413–423 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Khamis, M.A., Ngo, H.Q., Ré, C., Rudra, A.: Joins via geometric resolutions: worst case and beyond. ACM Trans. Database Syst. 41(4), 22:1–22:45 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kopelowitz, T., Pettie, S., Porat, E.: Dynamic set intersection. In: Dehne, F., Sack, J.-R., Stege, U. (eds.) WADS 2015. LNCS, vol. 9214, pp. 470–481. Springer, Cham (2015). doi: 10.1007/978-3-319-21840-3_39 CrossRefGoogle Scholar
  21. 21.
    Kopelowitz, T., Pettie, S., Porat, E.: Higher lower bounds from the 3SUM conjecture. In: Proceedings of the 27th SODA, pp. 1272–1287. SIAM (2016)Google Scholar
  22. 22.
    Lagraa, S., Seba, H.: An efficient exact algorithm for triangle listing in large graphs. Data Min. Knowl. Disc. 30(5), 1350–1369 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Latapy, M.: Main-memory triangle computations for very large (sparse (power-law)) graphs. Theor. Comput. Sci. 407(1–3), 458–473 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lee, T., Magniez, F., Santha, M.: Improved quantum query algorithms for triangle detection and associativity testing. Algorithmica 77(2), 459–486 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci. 20(2), 219–230 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mertzios, G.B., Nichterlein, A., Niedermeier, R.: The power of linear-time datareduction for maximum matching. In: Proceedings of the 42nd MFCS, LIPIcs, vol. 83, pp. 46:1–46:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017)Google Scholar
  27. 27.
    Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Park, H., Silvestri, F., Kang, U., Pagh, R.: Mapreduce triangle enumeration with guarantees. In: Proceedings of CIKM 2014, pp. 1739–1748. ACM (2014)Google Scholar
  29. 29.
    Patrascu, M.: Towards polynomial lower bounds for dynamic problems. In: Proceedings of the 42nd STOC, pp. 603–610. ACM (2010)Google Scholar
  30. 30.
    Schank, T., Wagner, D.: Finding, counting and listing all triangles in large graphs, an experimental study. In: Nikoletseas, S.E. (ed.) WEA 2005. LNCS, vol. 3503, pp. 606–609. Springer, Heidelberg (2005). doi: 10.1007/11427186_54 CrossRefGoogle Scholar
  31. 31.
    Sorge, M., Weller, M.: The graph parameter hierarchy, TU Berlin (2016). Unpublished ManuscriptGoogle Scholar
  32. 32.
    Zhang, Y., Parthasarathy, S.: Extracting analyzing and visualizing triangle \(k\)-core motifs within networks. In: Proceedings of the 28th ICDE, pp. 1049–1060. IEEE Computer Society (2012)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Matthias Bentert
    • 1
    Email author
  • Till Fluschnik
    • 1
  • André Nichterlein
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany

Personalised recommendations