We consider an online version of a longest path problem in an undirected and planar graph that is motivated by a location and routing problem occurring in the board game "Turn & Taxis". Path extensions have to be selected based on only partial knowledge on the order in which nodes become available in later iterations. Besides board games, online path extension problems have applications in disaster relief management when infrastructure has to be rebuilt after natural disasters. For example, flooding may affect large parts of a road network, and parts of the network may become available only iteratively and decisions may have to be made without the possibility of planning ahead. We suggest and analyse selection criteria that identify promising nodes (locations) for path extensions. We introduce the concept of tentacles of paths as an indicator for the future extendability. Different initialization and extension heuristics are suggested on compared to an ideal solution that is obtained by an integer linear programming formulation assuming complete knowledge, i.e., assuming that the complete sequence in which nodes become available is known beforehand. All algorithms are tested and evaluated on the original "Turn & Taxis" graph, and on an extended version of the "Turn & Taxis" graph, with different parameter settings. The numerical results confirm that the number of tentacles is a useful criterion when selecting path extensions, leading to near-optimal paths at relatively low computational costs.

翻译：我们考虑无向平面图中最长路径问题的在线版本，该问题源自棋盘游戏"Turn & Taxis"中的选址与路径规划场景。路径扩展必须基于对后续迭代中节点可用顺序的部分信息进行选择。除棋盘游戏外，在线路径扩展问题在自然灾害后基础设施重建的灾害应急管理中也具有应用价值。例如，洪水可能影响路网的大片区域，部分路段可能仅能迭代式恢复可用，且决策必须在无法提前规划的情况下做出。我们提出并分析了识别路径扩展候选节点（位置）的选择标准，引入路径触手（tentacle）概念作为未来可扩展性的指标。提出了不同的初始化和扩展启发式算法，并与基于整数线性规划建模的假设完全信息（即预先知道节点可用顺序的完整序列）的理想解进行对比。所有算法均在原始"Turn & Taxis"图及其扩展版本上，采用不同参数设置进行测试与评估。数值结果证实，在选择路径扩展时，触手数量是一个有效标准，能够以相对较低的计算成本获得接近最优的路径。