Advertisement

Algorithmica

, Volume 62, Issue 3–4, pp 637–658 | Cite as

On Independent Sets and Bicliques in Graphs

  • Serge GaspersEmail author
  • Dieter Kratsch
  • Mathieu Liedloff
Article

Abstract

Bicliques of graphs have been studied extensively, partially motivated by the large number of applications. In this paper we improve Prisner’s upper bound on the number of maximal bicliques (Combinatorica, 20, 109–117, 2000) and show that the maximum number of maximal bicliques in a graph on n vertices is Θ(3 n/3). Our major contribution is an exact exponential-time algorithm. This branching algorithm computes the number of distinct maximal independent sets in a graph in time O(1.3642 n ), where n is the number of vertices of the input graph. We use this counting algorithm and previously known algorithms for various other problems related to independent sets to derive algorithms for problems related to bicliques via polynomial-time reductions.

Keywords

Counting algorithms Combinatorial bound Maximal bicliques Maximal independent sets Exact exponential time algorithm 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alber, J., Niedermeier, R.: Improved tree decomposition based algorithms for domination-like problems. In: Proc. of LATIN 2002. LNCS, vol. 2286, pp. 613–627. Springer, Berlin (2002) CrossRefGoogle Scholar
  2. 2.
    Alexe, G., Alexe, S., Crama, Y., Foldes, S., Hammer, P.L., Simeone, B.: Consensus algorithms for the generation of all maximal bicliques. Discrete Appl. Math. 145, 11–21 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Amilhastre, J., Vilarem, M.C., Janssen, P.: Complexity of minimum biclique cover and minimum biclique decomposition for bipartite dominofree graphs. Discrete Appl. Math. 86, 125–144 (1998) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion–exclusion. SIAM J. Comput. 39, 546–563 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Dahllöf, V., Jonsson, P.: An algorithm for counting maximum weighted independent sets and its applications. In: Proc. of SODA 2002, pp. 292–298. ACM and SIAM, Philadelphia (2002) Google Scholar
  6. 6.
    Dahllöf, V., Jonsson, P., Wahlström, M.: Counting models for 2SAT and 3SAT formulae. Theor. Comput. Sci. 332, 265–291 (2005) CrossRefzbMATHGoogle Scholar
  7. 7.
    Dawande, M., Swaminathan, J., Keskinocak, P., Tayur, S.: On bipartite and multipartite clique problems. J. Algorithms 41, 388–403 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Demaine, E.D., Gutin, G., Marx, D., Stege, U.: Open Problems—Structure Theory and FPT Algorithmcs for Graphs, Digraphs and Hypergraphs. Dagstuhl Seminar Proceedings 07281 (2007), IBFI, Schloss Dagstuhl, Germany Google Scholar
  9. 9.
    Dias, V.M.F., Herrera de Figueiredo, C.M., Szwarcfiter, J.L.: Generating bicliques of a graph in lexicographic order. Theor. Comput. Sci. 337, 240–248 (2005) CrossRefzbMATHGoogle Scholar
  10. 10.
    Dias, V.M.F., Herrera de Figueiredo, C.M., Szwarcfiter, J.L.: On the generation of bicliques of a graph. Discrete Appl. Math. 155, 1826–1832 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Fernau, H., Kneis, J., Kratsch, D., Langer, A., Liedloff, M., Raible, D., Rossmanith, P.: An exact algorithm for the maximum leaf spanning tree problem. In: Proc. of IWPEC 2009. LNCS, vol. 5917, pp. 161–172. Springer, Berlin (2009) Google Scholar
  12. 12.
    Fomin, F.V., Gaspers, S., Pyatkin, A.V., Razgon, I.: On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Fomin, F.V., Grandoni, F., Kratsch, D.: A measure & conquer approach for the analysis of exact algorithms. J. ACM 56 (2009) Google Scholar
  14. 14.
    Fürer, M., Kasiviswanathan, S.P.: Algorithms for counting 2-SAT solutions and colorings with applications. In: Proc. of AAIM 2007. LNCS, vol. 4508, pp. 47–57. Springer, Berlin (2007) Google Scholar
  15. 15.
    Ganter, B., Wille, R.: Formal Concept Analysis, Mathematical Foundations. Springer, Berlin (1996) zbMATHGoogle Scholar
  16. 16.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, New York (1979) zbMATHGoogle Scholar
  17. 17.
    Gaspers, S., Liedloff, M.: A branch-and-reduce algorithm for finding a minimum independent dominating set. ArXiv Report 1009.1381 [CoRR abs] (2010)
  18. 18.
    Gaspers, S., Kratsch, D., Liedloff, M.: On independent sets and bicliques in graphs. In: Proc. of WG 2008. LNCS, vol. 5344, pp. 171–182. Springer, Berlin (2008) Google Scholar
  19. 19.
    Hochbaum, D.S.: Approximating clique and biclique problems. J. Algorithms 29, 174–200 (1998) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Kneis, J., Langer, A., Rossmanith, P.: A fine-grained analysis of a simple independent set algorithm. In: Proc. of FSTTCS 2009. LIPICS, vol. 4, pp. 287–298, Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Germany Google Scholar
  21. 21.
    Makino, K., Uno, T.: New algorithms for enumerating all maximal cliques. In: Proc. of SWAT 2004. LNCS, vol. 3111, pp. 260–272. Springer, Berlin (2004) Google Scholar
  22. 22.
    Moon, J.W., Moser, L.: On cliques in graphs. Isr. J. Math. 3, 23–28 (1965) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Nourine, L., Raynaud, O.: A fast algorithm for building lattices. Inf. Process. Lett. 71, 199–204 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Nourine, L., Raynaud, O.: A fast incremental algorithm for building lattices. J. Exp. Theor. Artif. Intell. 14, 217–227 (2002) CrossRefzbMATHGoogle Scholar
  25. 25.
    Okamoto, Y., Uno, T., Uehara, R.: Counting the number of independent sets in chordal graphs. J. Discrete Algorithms 6, 229–242 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Peeters, R.: The maximum edge biclique problem is NP-complete. Discrete Appl. Math. 131, 651–654 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Prisner, E.: Bicliques in graphs I: Bounds on their number. Combinatorica 20, 109–117 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Robson, J.M.: Algorithms for maximum independent sets. J. Algorithms 7, 425–440 (1986) CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    van Rooij, J.M.M., Bodlaender, H.L.: Design by measure and conquer: a faster exact algorithm for dominating set. In: Proc. of STACS 2008. LIPIcs, pp. 657–668. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Germany Google Scholar
  30. 30.
    van Rooij, J.M.M., Nederlof, J., van Dijk, T.C.: Inclusion/exclusion meets measure and conquer. In: Proc. of ESA 2009. LNCS, vol. 5757, pp. 554–565. Springer, Berlin (2009) CrossRefGoogle Scholar
  31. 31.
    Wahlström, M.: A tighter bound for counting max-weight solutions to 2SAT instances. In: Proc. of IWPEC 2008. LNCS, vol. 5018, pp. 202–213. Springer, Berlin (2008) Google Scholar
  32. 32.
    Yannakakis, M.: Node and edge deletion NP-complete problems. In: Proc. of STOC 1978, pp. 253–264. ACM, New York (1978) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Serge Gaspers
    • 1
    Email author
  • Dieter Kratsch
    • 2
  • Mathieu Liedloff
    • 3
  1. 1.Institute of Information SystemsVienna University of TechnologyViennaAustria
  2. 2.LITAUniversité Paul Verlaine-MetzMetz Cedex 01France
  3. 3.LIFOUniversité d’OrléansOrléans Cedex 2France

Personalised recommendations