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：我们研究了PSPACE完全的$k$-加拿大旅行者问题，其中给定一个加权图$G=(V,E,\omega)$，以及源点$s\in V$和目标点$t\in V$。该问题还包含一个隐藏输入$E_* \subsetneq E$，其基数至多为$k$，表示被阻塞的边。目标是从$s$到$t$以最小距离行进。在行程开始时，阻塞边集$E_*$是未知的：旅行者在访问某条边的端点时才发现该边被阻塞。针对此问题，已有在线算法（也称为策略）被提出，并通过竞争比（即旅行者实际行进距离与提前知晓阻塞情况时应行进距离的比值）进行评估。尽管即使在树宽为2的单位加权平面图上，最优竞争比也是$2k+1$，但我们设计了一种多项式时间策略，在单位加权外平面图上实现了竞争比$9$。该值$9$也是此类图族的下界：我们证明，对于任意$\varepsilon > 0$，没有任何策略能达到竞争比$9-\varepsilon$。最后，我们证明在加权外平面图上无法实现（独立于$G$和$k$的）常数竞争比。