Deriving Compact QUBO Models via Multilevel Constraint Transformation

Abstract: With the advances in customized hardware for quantum annealing and digital/CMOS Annealing, Quadratic Unconstrained Binary Optimization (QUBO) models have received growing attention in the optimization literature. Motivated by an existing general-purpose approach that derives QUBO models from binary linear programs (BLP), we propose a novel Multilevel Constraint Transformation Scheme (MLCTS) that derives QUBO models with fewer ancillary binary variables. We formulate sufficient conditions for the existence of a compact QUBO formulation (i.e., in the original BLP decision space) in terms of constraint levelness and demonstrate the flexibility and applicability of MLCTS on synthetic examples and several well-known combinatorial optimization problems, i.e., the Maximum 2-Satisfiability Problem, the Linear Ordering Problem, the Community Detection Problem, and the Maximum Independence Set Problem. For a proof-of-concept, we compare the performance of two QUBO models for the latter problem on both a general-purpose software-based solver and a hardware-based QUBO solver. The MLCTS-derived models demonstrate significantly better performance for both solvers, in particular, solving up to seven times more instances with the hardware-based approach.