Given a bipartite graph and the number bicliques K, how many ways exist to solve the biclique cover problem using K (possibly empty) bicliques?

  • 2
    $\begingroup$ What are you looking for, an algorithm that counts the number of biclique covers, or an upper/lower bound in terms of n and K? Are there any constraints that the cover should be minimal in some way? $\endgroup$
    – daniello
    Mar 21 '17 at 22:15
  • $\begingroup$ Ideally I am looking for a closed form solution (My hunch is that it must exist). There are no constraints on the cover being minimal. $\endgroup$
    – tammo
    Mar 22 '17 at 12:42
  • $\begingroup$ Echoing daniello's comment: Your answer "closed form solution" suggests that you are asking for an expression that counts the number of biclique covers with $K$ cliques. But this makes sense only if you have a particular bipartite graph in mind. Can you rephrase your question to clarify what you are asking? $\endgroup$
    – Neal Young
    Mar 24 '17 at 21:36
  • $\begingroup$ I am thinking about some parametrisation of all possible graphs. Essentially the number of vertices and some measure of heterogeneity of the vertex distribution should suffice. $\endgroup$
    – tammo
    Mar 28 '17 at 9:12

Given a Graph $G=(V, E),$ it takes $O(|V|^3)$ delay and $O(2^|V|)$space to find all bicliques in lexicographical order in $G$ [1]. There is no polynomial-delay enumeration algorithm for all bicliques in reverse lexicographical order unless P=NP [2].

Given a bipartite graph $G=(U, V, E),$ it takes $O((|U|+|V|)^2)$ delay with exponential space to find all maximal bicliques in $G$ in lexicographical order in $U$ [3].

[1] Vania M.F. Dias, Celina M.H. de Figueiredo, and Jayme L. Szwarcfiter. “Generating Bicliques of a Graph in Lexicographic Order”. In: Theoretical Computer Science 337.1-3 (June 2005), pp. 240–248. doi: 10.1016/j.tcs.2005.01.014.

[2] Wasa, Kunihiro. "Enumeration of enumeration algorithms." arXiv preprint arXiv:1605.05102 (2016).

[3] Alain Gely, Lhouari Nourine, and Bachir Sadi. “Enumeration aspects of maximal cliques and bicliques”. In: Discrete Applied Mathematics 157.7 (Apr. 2009), pp. 1447–1459. doi: 10.1016/j.dam.2008.10. 010.


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