The bipartite unconstrained 0–1 quadratic programming problem: Polynomially solvable cases

We consider the bipartite unconstrained 0–1 quadratic programming problem (BQP01) which is a generalization of the well studied unconstrained 0–1 quadratic programming problem (QP01). BQP01 has numerous applications and the problem is known to be MAX SNP hard. We show that if the rank of an associated m×nm×n cost matrix Q=(qij)Q=(qij) is fixed, then BQP01 can be solved in polynomial time. When QQ is of rank one, we provide an O(nlogn)O(nlogn) algorithm and this complexity reduces to O(n)O(n) with additional assumptions. Further, if qij=ai+bjqij=ai+bj for some aiai and bjbj, then BQP01 is shown to be solvable in O(mnlogn)O(mnlogn) time. By restricting m=O(logn)m=O(logn), we obtain yet another polynomially solvable case of BQP01 but the problem remains MAX SNP hard if m=O(nk) for a fixed kk. Finally, if the minimum number of rows and columns to be deleted from QQ to make the remaining matrix non-negative is O(logn)O(logn), then we show that BQP01 is polynomially solvable but it is NP-hard if this number is O(nk) for any fixed kk.