Biclique Sandwich: bounding the size of a maximum biclique

I. Basic Concepts

1. Bipartitie graph

A bipartite graph, is an undirected graph G = (V, E) in which V can be partitioned into two sets V1 and V2 such that (u, v) ∈ E implies either u ∈ V1 and v ∈ V2 or u ∈ V2 and v ∈ V1. That is, the set of vertices is decomposed into two disjoint sets such that no two vertices within the same set are adjacent.

2. Biclque

Given a bipartite graph B = (V1 ∪ V2, E), a biclique C = U1 ∪ U2 is a subset of the vertex set, such that U1 ⊆ V1, U2 ⊆ V2, and for every u ∈ U1, v ∈ U2, the edge (u, v) ∈ E. In other words, a biclique is a complete bipartite subgraph of B.

II. Problem definition

1. Maximum Edge Biclique Problem(MBP)

Given a bipartite graph B = (V1 ∪ V2, E), find the biclique with the maximum number of edges.

2. Hardness of the problem and related problems

A biclique C = U1 ∪ U2 clearly has |U1| + |U2| vertices and |U1| * |U2| edges. The computational complexity of deciding whether or not a bipartite graph contains a biclique of a certain size depends on what objective function is applied on the size of a biclique. The biclique problem with objective function max (|U1| + |U2|) is called Maximum Vertex Biclique problem. This problem can be solved in polynomial time because maximum vertex biclique is equivalent to the maximum independent set in bipartite graphs which, in turn, can be solved via the minimum cut algorithm. The biclique problem with objective function max |U1|, provided that |U1| = |U2|, is called Balanced Complete Bipartite Subgraph problem or Balanced Biclique problem. It is listed as GT24 among the NP-complete problems in Garey and Johnson's book. The Maximum Edge Biclique problem was proven to be NP-complete[1] by induction from 3SAT.

3. Bounding the size of the maximum edge biclique in bipartite graphs

Idea: Given that the MAX-CLIQUE is upper bounded by the Lovasz theta function, if a reduction from biclique to clique exists, we can bound the size of maximum edge biclique by applying Lovasz theta function on the transformed graph.

III. Reduction from Biclique to Clique

1. Reduction from biclique to clique

Let Γ = (V1 ∪ V2, E) be a bipartite graph. Define

LQ(Γ) := (E, {((u, v),(w, s)) | Γ(u, v, w, s) is a biclique});

in words, the vertices of LQ(Γ) are edges of Γ, two of them are adjacent if they intersect or lie in a common C4.

Note: A cycle on n vertices is denoted Cn

2. Lemma

Let Ω be a maximal clique in LQ(Γ). Then the subgraph

Γ_Ω := ∪_{(u,v) ∈ Ω}

Reference

1. Peeters, R. The maximum edge biclique problem is NP-complete. Tech. Rep. 789, Tilburg University: Faculty of Economics and Business administration, 2000.