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.

翻译：组合优化问题的近期进展强调了图神经网络（GNN）的潜力。受物理学启发的GNN（PI-GNN）求解器通过无监督学习寻找近似解，在大规模组合优化问题中受到广泛关注。然而，在贪心算法性能较差的相对稠密图组合优化问题上，PI-GNN求解器的表现尚未得到充分讨论。此外，由于PI-GNN求解器采用松弛策略，学习后需要将连续空间人工映射回原始离散空间，这可能削弱解的鲁棒性。本文通过数值实验证明，在稠密图的组合优化问题学习初期，PI-GNN求解器可能陷入所有变量均为零的局部解。为此，我们通过控制松弛变量的连续性与离散性来规避局部解并解决上述问题：（i）引入新的惩罚项，通过控制松弛变量的连续性与离散性消除局部解；（ii）提出新的连续松弛退火（CRA）策略。该退火策略优先利用连续性强化探索以规避局部解，再通过调度惩罚项优先获得离散解，直至松弛变量接近离散值，从而避免从连续空间向原始离散空间的人工映射。实验表明，在PI-GNN求解器难以找到合理解的稠密图组合优化问题以及相对稀疏图的问题上，本方法均获得更优结果。此外，计算时间扩展性与PI-GNN求解器保持一致。