Find all maximal complete bipartite subgraph from given bipartite graph

Question

Given is a bipartite graph, and we want to list all maximal complete bipartite sub-graph.

For instance,

vertex set L = {A, B, C, D}

vertex set R = {a, b, c, d, e}

edges: A-a, A-b, B-a, B-b, C-c, C-d, D-c, D-d, D-e

The maximal complete bipartite are:

{A,B}-{a, b}

{C,D}-{c, d}

{D} - {c, d, e}

I have found a brute force algorithm, O(2^n). I don't know if some approximation algorithm or randomized algorithm.

Answer 1

You can transform the problem to finding maximal cliques by adding edges between every pair of vertices in each part of the bipartite graph.

The Bron-Kerbosch algorithm can be used to list all maximal cliques in a graph (not necessarily bipartite). It is pretty easy to implement and has a slightly better worst-case time bound of O(3^(n/3)). There is also a fix-parameter tractable time bound in term of the degeneracy of the graph.