The Canadian Traveller Problem on outerplanar graphs

2024-03-04

International Symposium on Mathematical Foundations of Computer Science

AI Summary

This paper investigates the online path planning variant of the $k$-Canadian Traveller Problem ($k$-CTP) on unweighted outerplanar graphs: given an unweighted outerplanar graph $G$, a source $s$, and a destination $t$, at most $k$ edges are adversarially blocked (unknown in advance), and edge statuses are revealed only upon arrival at an endpoint; the objective is to minimize the competitive ratio—the worst-case ratio between the traversed path length and the optimal offline path length. We establish the first tight competitive ratio of $9$ for unweighted outerplanar graphs: we design a polynomial-time online algorithm achieving this bound and prove its optimality via a novel lower-bound construction. We further uncover an intrinsic connection to the cow-path problem. For weighted outerplanar graphs, we derive a strong asymptotic lower bound of $Omega(log k / log log k)$ using the Lambert $W$ function—significantly improving prior results. Our approach integrates graph-theoretic properties (outerplanarity, treewidth), competitive analysis, and combinatorial reduction techniques.

Technology Category

• Planning, Routing, and Scheduling: Planning under Uncertainty
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the $k$-Canadian Traveller Problem, where a weighted graph $G=(V,E,omega)$ with a source $sin V$ and a target $tin V$ 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 $frac{ln k}{ln ln k}$.

Problem

• Optimize travel path with hidden blockages
• Achieve minimal distance in outerplanar graphs
• Establish competitive ratio bounds for strategies

Innovation

• Polynomial-time strategy for outerplanar graphs
• Competitive ratio 9 achieved
• Lower bound analysis using Lambert W function