An Amortized Efficiency Threshold for Comparing Neural and Heuristic Solvers in Combinatorial Optimization

2026-05-14

AI Summary

This work addresses the lack of a fair energy-efficiency comparison framework between neural combinatorial optimizers and classical heuristics at deployment scale, despite concerns over the high training energy costs of neural solvers. We propose the Amortized Efficiency Threshold (AET)—the minimum number of problem instances required for a neural solver’s total energy consumption or carbon emissions to fall below those of a heuristic, under a solution-quality constraint. To enable rigorous evaluation, we establish the first unified protocol for assessing both operational energy use and carbon emissions, incorporating symmetric amortization of hardware manufacturing’s embodied carbon. Using the MTVRP benchmark with PyVRP’s HGS as the baseline heuristic and modeling GPU training and CPU inference energy costs, our experiments on n=20 instances yield an AET of approximately 1.58×10⁵, with a per-instance energy ratio of 0.41, demonstrating the energy efficiency advantage of neural solvers in large-scale deployments.

Abstract

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 Multi-Task VRP (MTVRP) environment at n=20 customers across 19 problem variants and five training seeds, with HGS via PyVRP as the heuristic baseline. The measured crossover sits near $1.58 \times 10^5$ deployed instances; the per-instance ratio is 0.41, reflecting the moderate size of the instances tested. The contribution is the framework, the open instrumentation, and the measurement protocol; structural convergence of the ratio at larger problem sizes is left to future empirical work.

Problem

• combinatorial optimization
• energy efficiency
• neural solvers
• heuristic solvers
• amortized cost

Innovation

• Amortized Efficiency Threshold
• neural combinatorial optimization
• energy efficiency
• carbon footprint
• heuristic solvers