Hybrid Annealing Method Based on subQUBO Model Extraction With Multiple Solution Instances

Abstract

Ising machines are expected to solve combinatorial optimization problems efficiently by representing them as Ising models or equivalent quadratic unconstrained binary optimization (QUBO) models . However, upper bound exists on the computable problem size due to the hardware limitations of Ising machines. This paper propose a new hybrid annealing method based on partial QUBO extraction, called subQUBO model extraction, with multiple solution instances. For a given QUBO model, the proposed method obtains <inline-formula><tex-math notation="LaTeX">$N_I$</tex-math></inline-formula> quasi-optimal solutions (quasi-ground-state solutions) in some way using a classical computer. The solutions giving these quasi-optimal solutions are called <i>solution instances</i> . We extract a size-limited subQUBO model as follows based on a strong theoretical background: we randomly select <inline-formula><tex-math notation="LaTeX">$N_S$</tex-math></inline-formula> <inline-formula><tex-math notation="LaTeX">$(N_S&lt;N_I)$</tex-math></inline-formula> solution instances among them and focus on a particular binary variable <inline-formula><tex-math notation="LaTeX">$x_i$</tex-math></inline-formula> in the <inline-formula><tex-math notation="LaTeX">$N_S$</tex-math></inline-formula> solution instances. If <inline-formula><tex-math notation="LaTeX">$x_i$</tex-math></inline-formula> value is much <i>varied</i> over <inline-formula><tex-math notation="LaTeX">$N_S$</tex-math></inline-formula> solution instances, it is included in the subQUBO model; otherwise, it is not. We find a (quasi-)ground-state solution of the extracted subQUBO model using an Ising machine and add it as a new solution instance. By repeating this process, we can finally obtain a (quasi-)ground-state solution of the original QUBO model. Experimental evaluations confirm that the proposed method can obtain better quasi-ground-state solution than existing methods for large-sized QUBO models.