Decision Diagrams (DDs) have emerged as a powerful tool for discrete optimization, with rapidly growing adoption. DDs are directed acyclic layered graphs; restricted DDs are a generalized greedy heuristic for finding feasible solutions, and relaxed DDs compute combinatorial relaxed bounds. There is substantial theory that leverages DD-based bounding, yet the complexity of constructing the DDs themselves has received little attention. Standard restricted DD construction requires $O(w \log(w))$ per layer; standard relaxed DD construction requires $O(w^2)$, where $w$ is the width of the DD. Increasing $w$ improves bound quality at the cost of more time and memory. We introduce implicit Decision Diagrams, storing arcs implicitly rather than explicitly, and reducing per-layer complexity to $O(w)$ for restricted and relaxed DDs. We prove this is optimal: any framework treating state-update and merge operations as black boxes cannot do better. Optimal complexity shifts the challenge from algorithmic overhead to low-level engineering. We show how implicit DDs can drive a MIP solver, and release ImplicitDDs.jl, an open-source Julia solver exploiting the implementation refinements our theory enables. Experiments demonstrate the solver outperforms Gurobi on Subset Sum.

翻译：决策图（DDs）已成为离散优化领域的重要工具，其应用范围正在迅速扩大。决策图是一种有向无环分层图：受限决策图是寻找可行解的广义贪心启发式方法，而松弛决策图则用于计算组合松弛边界。现有大量理论基于决策图边界方法，但决策图本身的构建复杂度却鲜少受到关注。标准受限决策图每层构建需要 $O(w \log(w))$ 复杂度；标准松弛决策图每层需要 $O(w^2)$ 复杂度，其中 $w$ 为决策图的宽度。增加 $w$ 可提升边界质量，但会消耗更多时间和内存。本文提出隐式决策图，通过隐式而非显式存储边，将受限与松弛决策图的每层复杂度降至 $O(w)$。我们证明该复杂度是最优的：任何将状态更新与合并操作视为黑盒的框架均无法取得更优结果。最优复杂度将挑战从算法开销转移至底层工程实现。我们展示了隐式决策图如何驱动混合整数规划求解器，并发布了开源Julia求解器ImplicitDDs.jl，该工具充分利用了我们理论所支持的实现优化。实验表明，该求解器在子集和问题上性能优于Gurobi。