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

TL;DR

This paper analyzes approximation algorithms for bipartite boolean quadratic programming, providing formulas for average solution values, dominance bounds, complexity results, and new rounding algorithms with practical performance insights.

Contribution

It introduces a closed-form formula for average solution value, establishes dominance bounds, and proposes new rounding algorithms with theoretical and computational validation.

Findings

Average solution value can be computed in O(mn) time.

Solutions with objective no worse than average dominate at least 2^{m+n-2} solutions.

A new integer programming formulation and rounding algorithms improve solution bounds.

Abstract

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 NP-hard. 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 a > 1, no polynomial time approximation algorithm exists for BBQP with dominance ratio larger than 1-2^{(m+n)(1-a)/a}, 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.