Abstract
Bicliques of graphs have been studied extensively, partially motivated by the large number of applications. One of the main algorithmic interests is in designing algorithms to enumerate all maximal bicliques of a (bipartite) graph. Polynomial-time reductions have been used explicitly or implicitly to design polynomial delay algorithms to enumerate all maximal bicliques.
Based on polynomial-time Turing reductions, various algorithmic problems on (maximal) bicliques can be studied by considering the related problem for (maximal) independent sets. In this line of research, we improve Prisner’s upper bound on the number of maximal bicliques [Combinatorica, 2000] and show that the maximum number of maximal bicliques in a graph on n vertices is exactly 3n/3 (up to a polynomial factor). The main results of this paper are O(1.3642n) time algorithms to compute the number of maximal independent sets and maximal bicliques in a graph.
A large part of the research was done while Serge Gaspers was visiting the University of Metz.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
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)
Dawande, M., Swaminathan, J., Keskinocak, P., Tayur, S.: On bipartite and multipartite clique problems. J. Algorithms 41, 388–403 (2001)
Dias, V.M.F., Herrera de Figueiredo, C.M., Szwarcfiter, J.L.: Generating bicliques of a graph in lexicographic order. Theoret. Comput. Sci. 337, 240–248 (2005)
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)
Fomin, F.V., Gaspers, S., Pyatkin, A.V., Razgon, I.: On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica, Special Issue on Parameterized and Exact Algorithms (to appear)
Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and conquer: Domination – A case study. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 192–203. Springer, Heidelberg (2005)
Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and conquer: A simple O(20.288n) independent set algorithm. In: SODA 2006, pp. 18–25. ACM Press, New York (2006)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A guide to the Theory of NP-completeness. Freeman, New York (1979)
Ganter, B., Wille, R.: Formal Concept Analysis, Mathematical Foundations. Springer, Berlin (1996)
Gaspers, S., Liedloff, M.: A branch-and-reduce algorithm for finding a minimum independent dominating set in graphs. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 78–89. Springer, Heidelberg (2006)
Hochbaum, D.S.: Approximating clique and biclique problems. J. Algorithms 29, 174–200 (1998)
Hujter, M., Tuza, Z.: The number of maximal independent sets in triangle-free graphs. SIAM J. Discrete Math. 6, 284–288 (1993)
Johnson, D.S., Papadimitriou, C.H.: On generating all maximal independent sets. Inf. Process. Lett. 27, 119–123 (1988)
Makino, K., Uno, T.: New algorithms for enumerating all maximal cliques. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 260–272. Springer, Heidelberg (2004)
Moon, J.W., Moser, L.: On cliques in graphs. Israel J. Math. 3, 23–28 (1965)
Nourine, L., Raynaud, O.: A Fast Algorithm for Building Lattices. Inf. Process. Lett. 71, 199–204 (1999)
Nourine, L., Raynaud, O.: A fast incremental algorithm for building lattices. J. Exp. Theor. Artif. Intell. 14, 217–227 (2002)
Okamoto, Y., Uno, T., Uehara, R.: Linear-Time Counting Algorithms for Independent Sets in Chordal Graphs. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 433–444. Springer, Heidelberg (2005)
Peeters, R.: The maximum edge biclique problem is NP-complete. Discrete Appl. Math. 131, 651–654 (2003)
Prisner, E.: Bicliques in Graphs I: Bounds on Their Number. Combinatorica 20, 109–117 (2000)
Robson, J.M.: Algorithms for maximum independent sets. J. Algorithms 7, 425–440 (1986)
van Rooij, J.M.M., Bodlaender, H.L.: Design by measure and conquer: a faster exact algorithm for dominating set. In: Albers, S., Weil, P. (eds.) STACS 2008. IBFI, Schloss Dagstuhl, Germany, pp. 657–668 (2008)
Wahlström, M.: A tighter bound for counting max-weight solutions to 2SAT instances. In: Grohe, M., Niedermeier, R. (eds.) IWPEC 2008. LNCS, vol. 5018, pp. 202–213. Springer, Heidelberg (2008)
Yannakakis, M.: Node and edge deletion NP-complete problems. In: STOC 1978, pp. 253–264. ACM, New York (1978)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gaspers, S., Kratsch, D., Liedloff, M. (2008). On Independent Sets and Bicliques in Graphs. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2008. Lecture Notes in Computer Science, vol 5344. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92248-3_16
Download citation
DOI: https://doi.org/10.1007/978-3-540-92248-3_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-92247-6
Online ISBN: 978-3-540-92248-3
eBook Packages: Computer ScienceComputer Science (R0)