AI 中文总结
该研究提出带有限制绕行的安全自行车网络（SBNBD）问题，分析其在多类图上的复杂度，证明其关于非安全边等参数的可处理性，并通过OpenStreetMap数据实验验证相关算法的有效性。
AI 中文摘要
我们提出了带有限制绕行的安全自行车网络（Safe Bicycle Network with Bounded Detours, SBNBD）问题，其研究动机是升级农村道路网络以满足自行车交通需求。给定一个包含安全边与非安全边、边长度、升级成本、终端对、预算以及绕行因子α的无向图，任务是对非安全边进行升级，使得每个终端对都能通过一条安全路径连接，且该路径长度不超过其在原始网络中最短路径长度的α倍。我们从参数化视角研究SBNBD问题。我们证明了该问题在受限图类上具有强NP-难解性，包括树宽为2的平面图、反馈顶点集数为1的图以及最大度为3的图；同时针对树和最大度为2的图，给出了多项式时间算法以补充这些下界。我们证明了该问题关于非安全边数量是固定参数可处理的，并证明了基于SETH的匹配下界、多项式核下界以及针对自然参数的W-难解性。我们的主要结构结果是将任意实例映射为等价实例，其顶点和边的数量为O(fes + p)，其中fes是反馈边数，p是终端对数量；这使得该问题关于fes + p是固定参数可处理的。最后，我们在OpenStreetMap道路网络（针对德国小型城市及其周边区域）上评估了基于ILP的算法。这些实例具有较小的树宽上界和适中的反馈边结构。基于fes + p归约的预处理以及基于树分解的割生成均提升了精确求解效果，尤其是在较难的实例上。对不同绕行因子的实验表明，增大α可减少升级边的长度，揭示了升级成本与允许的相对绕行之间的权衡。总体而言，图结构参数为安全自行车网络设计提供了一种有用的算法视角。
英文摘要
We introduce the \emph{Safe Bicycle Network with Bounded Detours} (\emph{SBNBD}) problem, motivated by upgrading rural road networks for bicycle traffic. Given an undirected graph with safe and unsafe edges, edge lengths, upgrade costs, terminal pairs, a budget, and a detour factor $α$, the task is to upgrade unsafe edges so that each terminal pair is connected by a safe path of length at most $α$ times its shortest-path distance in the original network. We study SBNBD from a parameterized perspective. We prove strong NP-hardness on restricted graph classes, including planar graphs of treewidth two, graphs with feedback vertex set number one, and graphs of maximum degree three, and complement these lower bounds with polynomial-time algorithms for trees and graphs of maximum degree two. We show fixed-parameter tractability for the number of unsafe edges and prove matching SETH-based lower bounds, a polynomial-kernel lower bound, and W-hardness for natural parameters. Our main structural result maps any instance to an equivalent instance with $O(\mathrm{fes}+p)$ vertices and edges, where $\mathrm{fes}$ is the feedback edge number and $p$ the number of terminal pairs; this yields fixed-parameter tractability for $\mathrm{fes}+p$. Finally, we evaluate ILP-based algorithms on OpenStreetMap road networks for small German municipalities and their surroundings. The instances have small treewidth upper bounds and moderate feedback edge structure. Preprocessing based on the $\mathrm{fes}+p$ reduction and tree-decomposition-based cut generation both improve exact solving, especially on harder instances. Experiments with different detour factors show that increasing $α$ can reduce the upgraded-edge length, revealing trade-offs between upgrade cost and allowed relative detours. Overall, structural graph parameters provide a useful algorithmic lens for safe bicycle-network design.
