The Linear Assignment Problem is a fundamental combinatorial optimization task where classical exact solvers ensure optimality but suffer from an $\mathcal{O}(N^{3})$ bottleneck, while recent neural approximations struggle with scalability and exactness. We propose a learning-augmented framework that accelerates exact solvers by predicting dual variables to warm-start the search, backed by a fallback mechanism to preserve worst-case guarantees. Central to our approach is RowDualNet, a lightweight, row-independent architecture that avoids the $\mathcal{O}(N^{2})$ memory bottleneck of graph models, enabling scalable neural warm-starting up to $N=16{,}384$. Feasibility is guaranteed by construction via the Min-Trick mechanism, completely eliminating the need for costly iterative projections. Empirically, our method drastically reduces the search effort of the Jonker-Volgenant (LAPJV) algorithm, yielding robust zero-shot generalization with strict optimality and end-to-end speedups of over 2x on complex synthetic data, 1.25x on real-world tracking, and 1.5x on transportation networks.

翻译：线性指派问题是组合优化中的基础问题，经典精确求解器虽能保证最优性却受限于 $\mathcal{O}(N^{3})$ 的计算瓶颈，而近期神经近似方法在可扩展性与精确性方面存在不足。我们提出一种学习增强框架，通过预测对偶变量加速精确求解器的搜索过程，并配备回退机制以保留最坏情况下的性能保证。该框架的核心是RowDualNet——一种轻量级行独立架构，避免了图模型常见的 $\mathcal{O}(N^{2})$ 内存瓶颈，支持对 $N=16{,}384$ 规模问题的可扩展神经热启动。通过最小化技巧机制保证可行性，完全消除了高成本迭代投影的需求。实验表明，本方法显著减少了Jonker-Volgenant (LAPJV)算法的搜索开销，在复杂合成数据上实现严格最优的零样本泛化与超过2倍的端到端加速，在真实追踪任务中达1.25倍加速，在运输网络中达1.5倍加速。