A Quantum Constraint Generation Framework for Binary Linear Programs
András Czégel, Boglárka G. -Tóth
TL;DR
Addressing binary linear programs with quantum optimization, the paper introduces a quantum-informed classical constraint-generation framework that relaxes the problem into a QUBO/Ising Hamiltonian, uses a quantum subroutine to sample potential solutions, and iteratively adds constraint coupling terms based on observed violations. It formalizes the Relaxed Quadratic Program (RQP) and Relaxed Problem Hamiltonian (RPH), defines a violation-score mechanism with a tunable threshold, and demonstrates on small weighted exact-set-cover instances that the approach yields higher feasibility and often better objective values than a plain quantum baseline. Theoretical results discuss the tradeoff between relaxation simplicity and Hamiltonian hardness, showing monotonic growth in Hamiltonian terms and improved feasibility across iterations, with proofs in the Appendix. Practically, the framework provides a modular, hardware-agnostic path to hybrid quantum-classical ILP solving that can scale with advances in quantum hardware and improvements in the underlying quantum subroutine.
Abstract
We propose a new approach to utilize quantum computers for binary linear programming (BLP), which can be extended to general integer linear programs (ILP). Quantum optimization algorithms, hybrid or quantum-only, are currently general purpose, standalone solvers for ILP. However, to consider them practically useful, we expect them to overperform the current state of the art classical solvers. That expectation is unfair to quantum algorithms: in classical ILP solvers, after many decades of evolution, many different algorithms work together as a robust machine to get the best result. This is the approach we would like to follow now with our quantum 'solver' solutions. In this study we wrap any suitable quantum optimization algorithm into a quantum informed classical constraint generation framework. First we relax our problem by dropping all constraints and encode it into an Ising Hamiltonian for the quantum optimization subroutine. Then, by sampling from the solution state of the subroutine, we obtain information about constraint violations in the initial problem, from which we decide which coupling terms we need to introduce to the Hamiltonian. The coupling terms correspond to the constraints of the initial binary linear program. Then we optimize over the new Hamiltonian again, until we reach a feasible solution, or other stopping conditions hold. Since one can decide how many constraints they add to the Hamiltonian in a single step, our algorithm is at least as efficient as the (hybrid) quantum optimization algorithm it wraps. We support our claim with results on small scale minimum cost exact cover problem instances.