Average value of solutions for the bipartite boolean quadratic programs and rounding algorithms

We consider domination analysis of approximation algorithms for the bipartite boolean quadratic programming problem (BBQP) with m+nm+n variables. A closed-form formula is developed to compute the average objective function value AA of all solutions in O(mn)O(mn) time. However, computing the median objective function value of the solutions is shown to be NP-hard. Also, we show that any solution with objective function value no worse than AA dominates at least 2m+n−22m+n−2 solutions and this bound is the best possible. Further, we show that such a solution can be identified in O(mn)O(mn) time and hence the domination ratio of this algorithm is at least 14. We then show that for any fixed natural numbers a and b   such that η=ab>1, no polynomial time approximation algorithm exists for BBQP with domination ratio larger than 1−2(1−η)η(m+n), unless P = NP. It is shown that some powerful local search algorithms can get trapped at a local maximum with objective function value less than AA. 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.