a b s t r a c tWe 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×n cost matrix Q = (q ij ) is fixed, then BQP01 can be solved in polynomial time. When Q is of rank one, we provide an O(n log n) algorithm and this complexity reduces to O(n) with additional assumptions. Further, if q ij = a i + b j for some a i and b j , then BQP01 is shown to be solvable in O(mn log n) time. By restricting m = O(log n), we obtain yet another polynomially solvable case of BQP01 but the problem remains MAX SNP hard if m = O( k √ n) for a fixed k. Finally, if the minimum number of rows and columns to be deleted from Q to make the remaining matrix nonnegative is O(log n), then we show that BQP01 is polynomially solvable but it is NP-hard if this number is O( k √ n) for any fixed k.
We consider domination analysis of approximation algorithms for the bipartite boolean quadratic programming problem (BBQP) with m + n variables. A closed form formula is developed to compute the average objective function value A of all solutions in O(mn) time. However, computing the median objective function value of the solutions is shown to be NPhard. Also, we show that any solution with objective function value no worse than A dominates at least 2 m+n−2 solutions and this bound is the best possible. Further, we show that such a solution can be identified in O(mn) time and hence the dominance ratio of this algorithm is at least 1 4 . We then show that for any fixed rational number α > 1, no polynomial time approximation algorithm exists for BBQP with dominance ratio larger than 1 − 2 (1−α) α (m+n) , unless P=NP. We then analyze some powerful local search algorithms and show that they can get trapped at a local maximum with objective function value less than A . One of our approximation algorithms has an interesting rounding property which provides a data dependent lower bound on the optimal objective function value. A new integer programming formulation of BBQP is also given and computational results with our rounding algorithms are reported.
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