Maximum Weighted Edge Biclique Problem on Bipartite Graphs

For a graph G, a complete bipartite subgraph of G is called a biclique of G. For a weighted graph where each edge a weight the Maximum Weighted Edge Biclique (MWEB) problem is to find a biclique H of G such that is maximum. The decision version of the MWEB problem is known to be NP-complete for bipartite graphs. In this paper, we show that the decision version of the MWEB problem remains NP-complete even if the input graph is a complete bipartite graph. On the positive side, if the weight of each edge is a positive real number in the input graph G, then we show that the MWEB problem is time solvable for bipartite permutation graphs, and time solvable for chain graphs, which is a subclass of bipartite permutation graphs.