Approximating Clique and Biclique Problems

We present here 2-approximation algorithms for several node deletion and edge deletion biclique problems and for an edge deletion clique problem. The biclique problem is to find a node induced subgraph that is bipartite and complete. The objective is to minimize the total weight of nodes or edges deleted so that the remaining subgraph is bipartite complete. Several variants of the biclique problem are studied here, where the problem is defined on bipartite graph or on general graphs with or without the requirement that each side of the bipartition forms an independent set. The maximum clique problem is formulated as maximizing the number (or weight) of edges in the complete subgraph. A 2-approximation algorithm is given for the minimum edge deletion version of this problem. The approximation algorithms given here are derived as a special case of an approximation technique devised for a class of formulations introduced by Hochbaum. All approximation algorithms described (and the polynomial algorithms for two versions of the node biclique problem) involve calls to a minimum cut algorithm. One conclusion of our analysis of the NP-hard problems here is that all of these problems are MAX SNP-hard and at least as difficult to approximate as the vertex cover problem. Another conclusion is that the problem of finding the minimum node cut-set, the removal of which leaves two cliques in the graph, is NP-hard and 2-approximable.

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