A Polynomial-Time Algorithm for Unconstrained Binary Quadratic Optimization

Abstract: In this paper, an exact algorithm in polynomial time is developed to solve unrestricted binary quadratic programs. The computational complexity is $O\left( n^{\frac{15}{2}}\right) $, although very conservative, it is sufficient to prove that this minimization problem is in the complexity class $P$. The implementation aspects are also described in detail with a special emphasis on the transformation of the quadratic program into a linear program that can be solved in polynomial time. The algorithm was implemented in MATLAB and checked by generating five million matrices of arbitrary dimensions up to 30 with random entries in the range $\left[ -50,50\right] $. All the experiments carried out have revealed that the method works correctly.