A Hybrid Quantum-Classical Heuristic to solve large-scale Integer Linear Programs

We present a method that integrates any quantum algorithm capable of finding solutions to integer linear programs into the Branch-and-Price algorithm, which is regularly used to solve large-scale integer linear programs with a specific structure. The role of the quantum algorithm is to find integer solutions to subproblems appearing in Branch-and-Price. Obtaining optimal or near-optimal integer solutions to these subproblems can increase the quality of solutions and reduce the depth and branching factor of the Branch-and-Price algorithm and hence reduce the overall running time. We investigate the viability of the approach by considering the Tail Assignment problem and the Quantum Approximate Optimization Algorithm (QAOA). Here, the master problem is the optimization problem Set Partitioning or its decision version Exact Cover and can be expressed as finding the ground state of an Ising spin glass Hamiltonian. For Exact Cover, our numerical results indicate that the required algorithm depth decreases with the number of feasible solutions for a given success probability of finding a feasible solution. For Set Partitioning, on the other hand, we find that for a given success probability of finding the optimal solution, the required algorithm depth can increase with the number of feasible solutions if the Hamiltonian is balanced poorly, which in the worst case is exponential in the problem size. We therefore address the importance of properly balancing the objective and constraint parts of the Hamiltonian. We empirically find that the approach is viable with QAOA if polynomial algorithm depth can be realized on quantum devices.