A common critique of neural combinatorial-optimization solvers is that they are less energy-efficient than CPU metaheuristics, given the operational energy cost of training them on GPUs. This paper examines the inferential step from "training is expensive" to "neural solvers are net-inefficient", which is where the critique actually goes wrong. Training the network costs a large fixed amount of GPU energy; running the metaheuristic costs a small amount of CPU energy on every instance, repeated as long as the solver is deployed. The two are not commensurable until a deployment volume is fixed. We define the Amortized Efficiency Threshold (AET) as the deployment volume above which a neural solver breaks even with a heuristic baseline in total energy or carbon, under an explicit constraint on solution quality. We show that the cumulative-energy ratio between the two solvers tends to a constant strictly below one whenever the network wins per instance, and that this limit does not depend on how the training cost was measured. An embodied-carbon term amortizes hardware fabrication symmetrically on both sides. We instantiate the framework on the CVRP environment at n=50 customers with the attention-based autoregressive solver of Kool et al. (2019), trained for 100 epochs on 20,000 instances over five random seeds, and HGS via PyVRP as the heuristic baseline. The measured operational crossover sits near 4.56e3 deployed instances at the median of a six-point baseline-budget sweep; the per-instance neural-to-heuristic ratio is 2.29e-3. The contribution is the framework, the open instrumentation, and the end-to-end measurement protocol. Code and benchmark pipeline are available at https://github.com/sohaibafifi/aet.

翻译：对神经组合优化求解器的一个常见批评是，它们的能效低于CPU元启发式算法，原因在于在GPU上训练它们需要消耗大量运行能量。本文考察了从“训练昂贵”到“神经求解器净效率低下”这一推论步骤，而这正是该批评的症结所在。训练神经网络消耗固定且大量的GPU能量；而运行元启发式算法则在每个实例上消耗少量CPU能量，并且只要求解器部署使用就会重复消耗。在确定部署量之前，这两者无法直接比较。我们将摊销效率阈值（AET）定义为：在解质量的明确约束下，神经求解器在总能耗或碳排放上与启发式基线持平所需的部署量。我们证明，只要神经网络在每个实例上获胜，两种求解器之间的累积能量比趋于严格小于1的常数，并且该极限不依赖于训练成本的度量方式。嵌入式碳项在两侧对称摊销了硬件制造。我们以n=50个客户的CVRP环境实例化了该框架，使用Kool等人（2019）提出的基于注意力的自回归求解器，在20,000个实例上训练100个epoch，采用五个随机种子，并以通过PyVRP实现的HGS作为启发式基线。实测的运行交叉点位于约4.56e3个已部署实例（在六点基线预算扫描的中位数处）；每个实例的神经比启发式能量比为2.29e-3。本文的贡献在于该框架、开源工具以及端到端测量协议。代码和基准测试管道可在https://github.com/sohaibafifi/aet获取。