Near Optimal Solutions for Maximum Quasi-bicliques

The maximum quasi-biclique problem has been proposed for finding interacting protein group pairs from large protein-protein interaction (PPI) networks. The problem is defined as follows: THE MAXIMUM QUASI-BICLIQUE PROBLEM: Given a bipartite graph G= (X∪ Y,E) and a number 0 < δ ≤ 0.5, find a subset Xopt of X and a subset Yopt of Y such that any vertex x ∈Xopt is incident to at least (1−δ)|Yopt | vertices in Yopt , any vertex y ∈ Yopt is incident to at least (1 − δ)|Xopt | vertices in Xopt and |Xopt | + |Yopt | is maximized. The problem was proved to be NP-hard. We design a polynomial time approximation scheme to give a quasi-biclique X′ ⊆ X and Y ′ ⊆ Y with |X′| + |Y ′| ≥ (1 − )(|Xopt | + |Yopt |) such that any vertex x ∈X′ is incident to at least (1 − δ − )|Y ′| vertices in Y ′ and any vertex y ∈ Y ′ is incident to at least (1 − δ − )|X′| vertices in X′ for any > 0, where Xopt and Yopt form the optimal solution.