Controlling Continuous Relaxation for Combinatorial Optimization

Recent advancements in combinatorial optimization (CO) problems emphasize the potential of graph neural networks (GNNs). The physics-inspired GNN (PI-GNN) solver, which finds approximate solutions through unsupervised learning, has attracted significant attention for large-scale CO problems. Nevertheless, there has been limited discussion on the performance of the PI-GNN solver for CO problems on relatively dense graphs where the performance of greedy algorithms worsens. In addition, since the PI-GNN solver employs a relaxation strategy, an artificial transformation from the continuous space back to the original discrete space is necessary after learning, potentially undermining the robustness of the solutions. This paper numerically demonstrates that the PI-GNN solver can be trapped in a local solution, where all variables are zero, in the early stage of learning for CO problems on the dense graphs. Then, we address these problems by controlling the continuity and discreteness of relaxed variables while avoiding the local solution: (i) introducing a new penalty term that controls the continuity and discreteness of the relaxed variables and eliminates the local solution; (ii) proposing a new continuous relaxation annealing (CRA) strategy. This new annealing first prioritizes continuous solutions and intensifies exploration by leveraging the continuity while avoiding the local solution and then schedules the penalty term for prioritizing a discrete solution until the relaxed variables are almost discrete values, which eliminates the need for an artificial transformation from the continuous to the original discrete space. Empirically, better results are obtained for CO problems on the dense graphs, where the PI-GNN solver struggles to find reasonable solutions, and for those on relatively sparse graphs. Furthermore, the computational time scaling is identical to that of the PI-GNN solver.