We study the PSPACE-complete $k$-Canadian Traveller Problem, where a weighted graph $G=(V,E,\omega)$ with a source $s\in V$ and a target $t\in V$ are given. This problem also has a hidden input $E_* \subsetneq E$ of cardinality at most $k$ representing blocked edges. The objective is to travel from $s$ to $t$ with the minimum distance. At the beginning of the walk, the blockages $E_*$ are unknown: the traveller discovers that an edge is blocked when visiting one of its endpoints. Online algorithms, also called strategies, have been proposed for this problem and assessed with the competitive ratio, i.e. the ratio between the distance actually traversed by the traveller divided by the distance we would have traversed knowing the blockages in advance. Even though the optimal competitive ratio is $2k+1$ even on unit-weighted planar graphs of treewidth 2, we design a polynomial-time strategy achieving competitive ratio $9$ on unit-weighted outerplanar graphs. This value $9$ also stands as a lower bound for this family of graphs as we prove that, for any $\varepsilon > 0$, no strategy can achieve a competitive ratio $9-\varepsilon$. Finally, we show that it is not possible to achieve a constant competitive ratio (independent of $G$ and $k$) on weighted outerplanar graphs.

翻译：我们研究PSPACE完全的$k$-加拿大旅行者问题，其中给定一个加权图$G=(V,E,\omega)$，包含源节点$s\in V$和目标节点$t\in V$。该问题还存在一个隐藏输入$E_* \subsetneq E$，其基数至多为$k$，表示被阻塞的边。目标是寻找从$s$到$t$的最短旅行距离。在行走开始时，阻塞边集$E_*$未知：旅行者仅在访问某条边的端点时才能发现该边被阻塞。针对该问题，已有研究提出在线算法（亦称策略），并通过竞争比进行评估，即旅行者实际行进距离与事先知晓阻塞边情况下所能行进距离的比值。尽管即使在树宽为2的单位权重平面图上，最优竞争比也为$2k+1$，但我们设计了一种多项式时间策略，在单位权重外平面图上实现了竞争比$9$。这一值$9$也是该类图的下界，因为我们证明：对任意$\varepsilon > 0$，不存在能实现竞争比$9-\varepsilon$的策略。最后，我们证明在加权外平面图上无法实现常数竞争比（独立于$G$和$k$）。