Bipartite Sandwiches

A semideenite relaxation (?) for the problem of nding the maximum number (?) of edges in a complete bipartite subgraph of a bipartite graph ? = (V1 V2; E) is considered. For a large class of graphs, the relaxation is better than the LP-relaxation described in 8]. It is shown that (?) is bounded from above by the Lovv asz theta function (LQ(?)) of the graph LQ(?) related to the line graph of ?. Using a construction similar to the strong product of graphs, it is shown that (?) < (?) < (LQ(?)) is possible. Moreover, there exists no constant C1 > 0 (resp. C2 > 0) satisfying C1(?) (?) (resp., C2(LQ(?)) (?)) for all ?. A technical tool, Kronecker product of semideenite programs (SDPs), is introduced and used extensively. It captures the situation when two SDPs S and S 0 determine, in a way, a third one, T , so that the optimum value opt(T) of T equals opt(S) opt(S 0). As a by-product, this technique is used to give a straightforward proof of the result that the theta function (?) of the strong (co)product of graphs ? equals (?)(().