The branch-and-bound algorithm based on decision diagrams introduced by Bergman et al. in 2016 is a framework for solving discrete optimization problems with a dynamic programming formulation. It works by compiling a series of bounded-width decision diagrams that can provide lower and upper bounds for any given subproblem. Eventually, every part of the search space will be either explored or pruned by the algorithm, thus proving optimality. This paper presents new ingredients to speed up the search by exploiting the structure of dynamic programming models. The key idea is to prevent the repeated exploration of nodes corresponding to the same dynamic programming states by storing and querying thresholds in a data structure called the Barrier. These thresholds are based on dominance relations between partial solutions previously found. They can be further strengthened by integrating the filtering techniques introduced by Gillard et al. in 2021. Computational experiments show that the pruning brought by the Barrier allows to significantly reduce the number of nodes expanded by the algorithm. This results in more benchmark instances of difficult optimization problems being solved in less time while using narrower decision diagrams.

翻译：Bergman等人于2016年提出的基于决策图的分支定界算法是一种通过动态规划公式求解离散优化问题的框架。该算法通过编译一系列有界宽度的决策图，为任意给定子问题提供下界与上界。最终，搜索空间中的每个部分都将被该算法探索或剪枝，从而证明最优性。本文提出了新的改进方法，通过利用动态规划模型的结构来加速搜索过程。其核心思想是存储并查询一种名为"屏障"的数据结构中的阈值，以避免重复探索对应相同动态规划状态的节点。这些阈值基于先前发现的偏解之间的支配关系，且可通过整合Gillard等人于2021年引入的过滤技术进一步增强。计算实验表明，屏障带来的剪枝能力显著减少了算法扩展的节点数量。这使得更多困难优化问题的基准实例能够在更短时间内求解，同时使用更窄的决策图即可完成。