🤖 AI Summary
Combinatorial optimization problems are notoriously difficult to solve due to their exponentially growing solution spaces, despite the fact that feasibility can be verified in polynomial time. This work proposes Neural Certificate Pricing (NCP), a method that trains a neural network within an unsupervised learning framework to predict dual prices and employs a structured recovery layer to generate primal solutions. NCP is the first approach to embed certificate consistency directly into the neural architecture, enabling amortized separation. Theoretically, it establishes that first-order errors in price prediction induce only second-order degradation in objective value, thereby guaranteeing solution quality. Empirical results demonstrate that NCP matches or significantly outperforms existing neural baselines across three classes of combinatorial optimization problems, achieving higher computational efficiency and stronger out-of-distribution generalization.
📝 Abstract
Combinatorial optimization (CO) problems are difficult because certifiable discrete structure induces exponential search. One needs to search over the set exponentially many candidates to certify optimality, however, the structural feasibility of a path, packing, or cover can be verified in polynomial time once supplied. In this study, we introduce Neural Certificate Pricing (NCP) that exploits this asymmetry under an unsupervised learning framework. A neural network is trained to predict certificate-level dual prices, while a structured recovery layer constructs the induced primal marginal. NCP can be viewed as amortized separation: instead of enumerating violated inequalities, it learns the residual prices through which their aggregate effect enters recovery. When the certificate-consistency condition holds, the recovered marginal is globally feasible, and a local theory shows that first-order errors in the predicted price induce only second-order loss in objective value. Across three classes of CO problems, NCP either outperforms state-of-the-art neural baselines by large margins or matches them at a fraction of the computation time, and shows stronger out-of-distribution generalization.