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The Canadian Traveller Problem on Outerplanar Graphs

Authors Laurent Beaudou , Pierre Bergé , Vsevolod Chernyshev , Antoine Dailly, Yan Gerard , Aurélie Lagoutte , Vincent Limouzy , Lucas Pastor

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ACM Subject Classification
  • Theory of computation → Online algorithms
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph algorithms
Keywords
  • Canadian Traveller Problem
  • Online algorithms
  • Competitive analysis
  • Outerplanar graphs

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Abstract

We study the k-Canadian Traveller Problem, where a weighted graph G = (V,E,ω) with a source s ∈ V and a target t ∈ V are given. This problem also has a hidden input E_* ⊊ 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 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 ε > 0, no strategy can achieve a competitive ratio 9-ε. Finally, we show that it is not possible to achieve a constant competitive ratio (independent of G and k) on weighted outerplanar graphs.

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Laurent Beaudou, Pierre Bergé, Vsevolod Chernyshev, Antoine Dailly, Yan Gerard, Aurélie Lagoutte, Vincent Limouzy, and Lucas Pastor. The Canadian Traveller Problem on Outerplanar Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.19

Author Details

Laurent Beaudou
  • Université Clermont-Auvergne, CNRS, Mines de Saint-Etienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-Ferrand, France
Pierre Bergé
  • Université Clermont-Auvergne, CNRS, Mines de Saint-Etienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-Ferrand, France
Vsevolod Chernyshev
  • Université Clermont-Auvergne, CNRS, Mines de Saint-Etienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-Ferrand, France
Antoine Dailly
  • Université Clermont-Auvergne, CNRS, Mines de Saint-Etienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-Ferrand, France
Yan Gerard
  • Université Clermont-Auvergne, CNRS, Mines de Saint-Etienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-Ferrand, France
Aurélie Lagoutte
  • Univ. Grenoble Alpes, CNRS, Grenoble INP, G-SCOP, 38000 Grenoble, France
Vincent Limouzy
  • Université Clermont-Auvergne, CNRS, Mines de Saint-Etienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-Ferrand, France
Lucas Pastor
  • Université Clermont-Auvergne, CNRS, Mines de Saint-Etienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-Ferrand, France

Funding

This work was supported by the International Research Center "Innovation Transportation and Production Systems" of the I-SITE CAP 20-25 and by the ANR project GRALMECO (ANR-21-CE48-0004).

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