The Bipartite Boolean Quadric Polytope

We consider the Bipartite Boolean Quadratic Programming Problem (BQP01), which generalizes the well-known Boolean quadratic programming problem (QP01). The model has applications in graph theory, matrix factorization and bioinformatics, among others. The primary focus of this paper is on studying the structure of the Bipartite Boolean Quadric Polytope (BQP${}^{m,n}$) resulting from a linearization of a quadratic integer programming formulation of BQP01.

We present some basic properties and partial relaxations of BQP${}^{m,n}$, as well as some families of facets and valid inequalities. We find facet-defining inequalities including a family of odd-cycle inequalities. We discuss various approaches to obtain a valid inequality and facets from those of the related Boolean quadric polytope. The key strategy is based on rounding coefficients, and it is applied to the families of clique and cut inequalities in BQP${}^{m,n}$.