Bipartite sandwiches: bounding the size of a maximum biclique

A semidefinite relaxation σ(Γ) for the problem of finding the maximum number κ(Γ) of edges in a complete bipartite subgraph of a bipartite graph Γ = (V1 ∪ V2, E) is considered. For a large class of graphs, the relaxation is better than the LP-relaxation described in [9]. It is shown that σ(Γ) is bounded from above by the Lovász theta function θ(LQ(Γ)) of the graph LQ(Γ) related to the line graph of Γ. Using a construction similar to the strong product of graphs, it is shown that κ(Γ) < σ(Γ) < θ(LQ(Γ)) is possible. Moreover, there exists no constant C1 > 0 (resp. C2 > 0) satisfying C1σ(Γ) ≤ κ(Γ) (resp., C2θ(LQ(Γ)) ≤ σ(Γ)) for all Γ. A technical tool, Kronecker product of semidefinite programs (SDPs), is introduced and used extensively. It captures the situation when two SDPs S and S′ determine, in a way, a third one, T , so that the optimum value opt(T ) of T equals opt(S) opt(S′). As a by-product, this technique is used to give a straightforward proof of the result that the theta function θ(Γ ·∆) of the strong (co)product of graphs Γ ·∆ equals θ(Γ)θ(∆).