We present a fast, scalable, data-driven approach for solving relaxations of 0-1 integer linear programs. We use a combination of graph neural networks (GNN) and the Lagrange decomposition based algorithm FastDOG (Abbas and Swoboda 2022b). We make the latter differentiable for end-to-end training and use GNNs to predict its algorithmic parameters. This allows to retain the algorithm's theoretical properties including dual feasibility and guaranteed non-decrease in the lower bound while improving it via training. We overcome suboptimal fixed points of the basic solver by additional non-parametric GNN update steps maintaining dual feasibility. For training we use an unsupervised loss. We train on smaller problems and test on larger ones showing strong generalization performance with a GNN comprising only around $10k$ parameters. Our solver achieves significantly faster performance and better dual objectives than its non-learned version, achieving close to optimal objective values of LP relaxations of very large structured prediction problems and on selected combinatorial ones. In particular, we achieve better objective values than specialized approximate solvers for specific problem classes while retaining their efficiency. Our solver has better any-time performance over a large time period compared to a commercial solver. Code available at https://github.com/LPMP/BDD

翻译：我们提出了一种快速、可扩展、数据驱动的方法，用于求解0-1整数线性规划的松弛问题。该方法结合了图神经网络（GNN）与基于拉格朗日分解的算法FastDOG（Abbas和Swoboda 2022b）。我们对后者进行了可微化处理以实现端到端训练，并利用GNN预测其算法参数。这使得在保持算法理论性质（包括对偶可行性及下界保证非下降）的同时，能够通过训练提升性能。针对基本求解器的次优不动点问题，我们通过额外的非参数化GNN更新步骤加以克服，同时维持对偶可行性。训练采用无监督损失函数，在较小问题上进行训练并测试于更大规模问题，展现出强大的泛化性能，其中GNN仅包含约1万个参数。相比非学习版本，我们的求解器在性能上显著提升，对偶目标值更优，对超大规模结构化预测问题及特定组合优化问题的LP松弛达到了接近最优的目标值。特别是在保持效率的同时，我们针对特定问题类别取得了优于专用近似求解器的目标值。相较于商业求解器，我们的求解器在长时间段内具有更优的任意时间性能。代码见https://github.com/LPMP/BDD。