Continuous Parallel Relaxation for Finding Diverse Solutions in Combinatorial Optimization Problems

Finding the optimal solution is often the primary goal in combinatorial optimization (CO). However, real-world applications frequently require diverse solutions rather than a single optimum, particularly in two key scenarios. The first scenario occurs in real-world applications where strictly enforcing every constraint is neither necessary nor desirable. Allowing minor constraint violations can often lead to more cost-effective solutions. This is typically achieved by incorporating the constraints as penalty terms in the objective function, which requires careful tuning of penalty parameters. The second scenario involves cases where CO formulations tend to oversimplify complex real-world factors, such as domain knowledge, implicit trade-offs, or ethical considerations. To address these challenges, generating (i) penalty-diversified solutions by varying penalty intensities and (ii) variation-diversified solutions with distinct structural characteristics provides valuable insights, enabling practitioners to post-select the most suitable solution for their specific needs. However, efficiently discovering these diverse solutions is more challenging than finding a single optimal one. This study introduces Continual Parallel Relaxation Annealing (CPRA), a computationally efficient framework for unsupervised-learning (UL)-based CO solvers that generates diverse solutions within a single training run. CPRA leverages representation learning and parallelization to automatically discover shared representations, substantially accelerating the search for these diverse solutions. Numerical experiments demonstrate that CPRA outperforms existing UL-based solvers in generating these diverse solutions while significantly reducing computational costs.