The Canadian Traveller Problem on outerplanar graphs

We study the $k$-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, {\em i.e.}, the ratio between the distance actually traversed by the traveller divided by the distance he 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 $9-\varepsilon$ on it. This comes actually from a strong connexion with another well-known online problem called the cow-path problem. Finally, we show that it is not possible to achieve a competitive ratio $e^{W(\frac{\ln k}{2})} - 1$ on arbitrarily weighted outerplanar graphs, where $W$ is the Lambert W function. This lower bound is asymptotically greater than $\frac{\ln k}{\ln \ln k}$.