带邻域的固定拓扑最小长度树
Fixed Topology Minimum-Length Trees with Neighborhoods
AI总结：
研究提出带邻域的固定拓扑最小长度树问题，并针对连续空间与网络嵌入分别设计MINLP、MILP及降维、数据驱动建网方法，实验验证了其有效性。
摘要：
本文提出带邻域的固定拓扑最小长度树问题，其目标是将一棵有根树状图嵌入到一个 $d$ 维度量空间中，同时在节点必须嵌入到某些受限区域的条件下，使其总长度最小。该问题在工程设计中高效布设电缆或管线路由方面具有重要应用。我们基于嵌入所在的定义域，提出了新颖的数学优化方法来求解该问题的不同版本。在嵌入映射到连续空间的情形下，我们给出了若干混合整数非线性优化公式。如果嵌入目标是一个网络，我们推导了一个混合整数线性规划公式，以及一种降维方法，该方法能够以更少的 CPU 时间求解更大规模的问题。我们还提出了一种数据驱动方法，用于根据问题实例构建合适的网络。我们报告了一系列计算实验的结果，这些结果验证了我们所提方案。
英文摘要：
In this paper, we introduce the Fixed Topology Minimum-Length Tree with Neighborhood Problem, which aims to embed a rooted tree-shaped graph into a $d$-dimensional metric space while minimizing its total length provided that the nodes must be embedded to some restricted areas. This problem has significant applications in efficiently routing cables or pipelines in engineering designs. We propose novel mathematical optimization-based approaches to solve different versions of the problem based on the domain for the embedding. In cases where the embedding maps to a continuous space, we provide several Mixed Integer Nonlinear Optimization formulations. If the embedding is to a network, we derive a mixed integer linear programming formulation as well as a dimensionality reduction methodology that allows for solving larger problems in less CPU time. A data-driven methodology is also proposed to construct a proper network based on the instance of the problem. We report the results of a battery of computational experiments that validate our proposal.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
