Finding a high-quality feasible solution to a combinatorial optimization (CO) problem in a limited time is challenging due to its discrete nature. Recently, there has been an increasing number of machine learning (ML) methods for addressing CO problems. Neural diving (ND) is one of the learning-based approaches to generating partial discrete variable assignments in Mixed Integer Programs (MIP), a framework for modeling CO problems. However, a major drawback of ND is a large discrepancy between the ML and MIP objectives, i.e., variable value classification accuracy over primal bound. Our study investigates that a specific range of variable assignment rates (coverage) yields high-quality feasible solutions, where we suggest optimizing the coverage bridges the gap between the learning and MIP objectives. Consequently, we introduce a post-hoc method and a learning-based approach for optimizing the coverage. A key idea of our approach is to jointly learn to restrict the coverage search space and to predict the coverage in the learned search space. Experimental results demonstrate that learning a deep neural network to estimate the coverage for finding high-quality feasible solutions achieves state-of-the-art performance in NeurIPS ML4CO datasets. In particular, our method shows outstanding performance in the workload apportionment dataset, achieving the optimality gap of 0.45%, a ten-fold improvement over SCIP within the one-minute time limit.

翻译：在有限时间内为组合优化（CO）问题寻找高质量的可行解因其离散性质而极具挑战性。近年来，越来越多的机器学习（ML）方法被用于解决CO问题。神经潜水（ND）是其中一种基于学习的方法，用于生成混合整数规划（MIP，一种建模CO问题的框架）中的部分离散变量赋值。然而，ND的一个主要缺点是ML与MIP目标之间存在巨大差异，即变量值分类精度相对于原始界限的偏差。我们的研究发现，特定范围的变量赋值率（覆盖率）能产生高质量的可行解，并建议优化覆盖率可弥合学习目标与MIP目标之间的差距。因此，我们提出了一种事后方法及一种基于学习的覆盖优化方法。该方法的核心思想是联合学习限制覆盖搜索空间，并在学习到的搜索空间中预测覆盖率。实验结果表明，通过学习深度神经网络估计覆盖率以寻找高质量可行解，在NeurIPS ML4CO数据集上达到了最先进性能。特别地，我们的方法在工作负载分配数据集上表现卓越，实现了0.45%的最优性差距，在一分钟时间限制内相较于SCIP提升了十倍。