DualCert：带约束耦合学习的旅行商问题求解器
DualCert: A Solver for the Traveling Salesman Problem with Constraint-Coupled Learning
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中文总结 AI 辅助
DualCert是带约束耦合学习的TSP求解器，通过引入约束耦合学习结合确定性验证，在TSP1000实例上实现了优于NeuroLKH的性能，保证了输出有效性。
中文摘要 AI 辅助
大规模旅行商问题（TSP）实例需要求解器在分配有限计算资源的同时保证输出的有效性。现有的神经-运筹学（OR）混合模型在预测指导时，未要求学习到的转移满足搜索过程中发现的约束。DualCert引入了**约束耦合学习**，其中当前的度方程与动态分离的子图消除约束（SECs）定义了每个学习到的转移。在每次优化迭代中，度方程和选定的、严格满足的带有正松弛量的SEC方程，共同定义了依赖于迭代点的原-松弛卡鲁什-库恩-塔克（KKT）流形。修复后的对偶变量与被违反的SEC行定义了局部代价场。精确的约束镜像下降步骤将每个有限状态映射到同一流形上的正状态。在选定行和确定性关联保持固定的情况下，隐式微分将参数扰动映射到流形切空间，并复用前向约束算子计算局部代价场的导数。终端边状态在固定预算下，在Held-Karp上升、候选图边测试和巡游构建之间分配计算资源。确定性验证会重新计算原始代价，仅接受经过验证的候选图下界和边决策。在1000个保留的TSP1000实例上，DualCert在每个实例9.55秒的批量摊销时间内，相对于Lin-Kernighan-Helsgaun第3版（LKH-3）参考巡游，实现了0.0573%的平均巡游代价差距，为每个实例返回了经过验证的候选图下界，并达到了81.46%的边决策覆盖率，其平均差距比已报道的NeuroLKH平均差距小67.1%。因此，优化约束主导学习过程，而确定性验证则保证了输出的有效性。
英文摘要
Large traveling salesman problem (TSP) instances require a solver to allocate limited computation while preserving the validity of its outputs. Existing neural--operations-research (OR) hybrids predict guidance without requiring learned transitions to satisfy constraints discovered during search. DualCert introduces \emph{constraint-coupled learning}, in which current degree equations and dynamically separated subtour-elimination constraints (SECs) define each learned transition. At each refinement, the degree equations and selected, strictly satisfied SEC equations, with positive slacks, define an iterate-dependent primal-slack Karush--Kuhn--Tucker (KKT) manifold. Repaired dual variables and violated SEC rows define a local cost field. An exact constrained mirror-descent step maps each finite state to a positive state on the same manifold. Where selected rows and deterministic ties remain fixed, implicit differentiation maps parameter perturbations into the manifold tangent space and reuses the forward constraint operator for the local-cost-field derivative. The terminal edge state allocates computation across Held--Karp ascent, candidate-graph edge tests, and tour construction under a fixed budget. Deterministic verification recomputes original costs and accepts only verified candidate-graph lower bounds and edge decisions. On 1,000 held-out TSP1000 instances, DualCert attains a mean tour-cost gap of \(0.0573\%\) from Lin--Kernighan--Helsgaun version 3 (LKH-3) reference tours in \(9.55\) batch-amortized seconds per instance. It returns a verified candidate-graph lower bound for every instance and achieves \(81.46\%\) edge-decision coverage. The mean gap is \(67.1\%\) smaller than the reported NeuroLKH mean gap. Thus, optimization constraints govern learning, while deterministic verification preserves output validity.
