连续时间服务网络设计中的动态离散化发现最优细化

中文摘要 AI 辅助

动态离散化发现（DDD）是一种通过迭代求解和细化问题松弛版本来解决问题的框架。DDD 是多种问题最先进算法的基础，包括我们在此研究的连续时间服务网络设计问题（CTSNDP）。在细化步骤中，此类算法必须识别松弛解中导致其对于原始问题不可行的冲突，然后对松弛应用修复以防止冲突再次发生。我们研究了最优细化问题，即从一组修复中选择一个子集，该子集修复所有冲突并产生最小规模的松弛，目的是使其更易于求解。我们证明，虽然冲突数量可能随实例参数呈指数增长，但我们可以更紧凑地表示它们，仅使用伪多项式空间和时间。基于我们更紧凑的表示，我们开发了求解细化问题的精确方法。最佳方法使我们能够在可忽略的时间内找到小规模松弛，对于某些实例类别，比自然的整数规划公式快 100 倍以上。在 CTSNDP 的 DDD 算法中使用这种细化方法，我们观察到与现有方法相比，模型规模和运行时间减少了约 10% 至 20%。

英文摘要

Dynamic Discretisation Discovery (DDD) is a framework for solving a problem by iteratively solving and refining a relaxed version of the problem. DDD is the basis for state-of-the-art algorithms for various problems, including the continuous-time service network design problem (CTSNDP), which we study here. In the refinement step, such algorithms must identify conflicts in the relaxation solution that make it infeasible for the original problem and then apply fixes to the relaxation to prevent them from recurring. We study the optimal refinement problem of selecting from a set of fixes a subset that fixes all conflicts and yields a relaxation of minimum size, with the aim of making it easier to solve. We demonstrate that while the number of conflicts may be exponential in the instance parameters, we can represent them more compactly, using only pseudo-polynomial space and time. Based on our more compact representation, we develop exact methods for solving the refinement problem. The best method allows us to find small relaxations in negligible amounts of time, more than 100 times faster than a natural integer programming formulation for some instance classes. Using this refinement approach in a DDD algorithm for the CTSNDP, we observe reductions in model sizes and running times of about 10 to 20 % compared to existing approaches.

发表机构

• Chair of Operations Research, RWTH Aachen University（运筹学教席，亚琛工业大学）
• RWTH Aachen University（亚琛工业大学）

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