The Canadian Traveller Problem on outerplanar graphs

TL;DR

This work investigates the $k$-Canadian Traveller Problem on outerplanar graphs, addressing the gap between the general $2k+1$ competitive bound and improved performance in restricted graph classes. It introduces ExpBalancing, a polynomial-time strategy that attains a constant $9$-competitive ratio on unit-weighted outerplanar graphs and proves this bound is tight by relating the problem to the classical linear search (cow-path) problem on shell graphs. It further proves that no constant competitive ratio is possible for arbitrarily weighted outerplanar graphs, establishing a lower bound $g(k)=e^{W\left(\frac{\ln k}{2}\right)}-1$, which grows sublinearly and satisfies $\frac{\ln k}{\ln \ln k} \le g(k) \le \ln k$. By connecting online blockage-aware routing to classic search problems, the paper delineates the limits of constant-competitive strategies and proposes directions toward $O(\log k)$ bounds and broader graph families.

Abstract

We study the $k$-Canadian Traveller Problem, where a weighted graph $G=(V,E,ω)$ 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 $\frac{\ln k}{\ln \ln k}$.

Figures (9)

• Figure 1: Example of an outerplanar graph: $p_{2}q_\ell$, $p_2q_{\ell-1}$, $p_3q_{\ell-2}$, and $p_{h-1}q_{4}$ are vertical chords and $q_1q_3$, $q_1q_4$ are horizontal chords.
• Figure 2: Westphal graph with $k=4$, as defined in We08
• Figure 3: Decomposing the graph into components with no articulation points and removing the useless components (the vertices in a dashed rectangle are the same in the original graph).
• Figure 6: Application of ExpBalancing on the first graph of the decomposition of \ref{['fig-unweightedOuterplanarGraphs-1-decomposition']}. At each step, the circled vertex is the one we are currently exploring, and we know the status of the bold edges: black is open, red is blocked.
• Figure 7: Application of ExpBalancing on the third graph of the decomposition of \ref{['fig-unweightedOuterplanarGraphs-1-decomposition']}. At each step, the circled vertex is the one we are currently exploring, and we know the status of the bold edges: black is open, red is blocked.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 4: Road maps
• Definition 5: $k$-CTP
• Lemma 6
• proof
• Lemma 7
• proof
• Lemma 8