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Improved Deterministic Strategy for the Canadian Traveller Problem Exploiting Small Max-(st)-Cuts

  • Pierre BergéEmail author
  • Lou Salaün
Conference paper
  • 83 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11926)

Abstract

The k-Canadian Traveller Problem consists in finding the optimal way from a source s to a target t on an undirected weighted graph G, knowing that at most k edges are blocked. The traveller, guided by a strategy, sees an edge is blocked when he visits one of its endpoints. A major result established by Westphal is that the competitive ratio of any deterministic strategy for this problem is at least \(2k+1\). reposition and comparison strategies achieve this bound.

We refine this analysis by focusing on graphs with a maximum (st)-cut size \(\mu _{\text {max}}\) less than k. A strategy called detour is proposed and its competitive ratio is \(2\mu _{\text {max}}+ \sqrt{2}(k-\mu _{\text {max}}) + 1\) when \(\mu _{\text {max}}< k\) which is strictly less than \(2k+1\). Moreover, when \(\mu _{\text {max}}\ge k\), the competitive ratio of detour is \(2k+1\) and is optimal. Therefore, detour improves the competitiveness of the deterministic strategies known up to now.

Keywords

Canadian traveller problem Competitive analysis Online algorithms 

References

  1. 1.
    Bar-Noy, A., Schieber, B.: The Canadian traveller problem. In: Proceedings of ACM/SIAM SODA, pp. 261–270 (1991)Google Scholar
  2. 2.
    Bender, M., Westphal, S.: An optimal randomized online algorithm for the k-Canadian Traveller Problem on node-disjoint paths. J. Comb. Optim. 30(1), 87–96 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bergé, P., Hemery, J., Rimmel, A., Tomasik, J.: On the competitiveness of memoryless strategies for the k-Canadian Traveller Problem. In: Kim, D., Uma, R.N., Zelikovsky, A. (eds.) COCOA 2018. LNCS, vol. 11346, pp. 566–576. Springer, Cham (2018).  https://doi.org/10.1007/978-3-030-04651-4_38CrossRefGoogle Scholar
  4. 4.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  5. 5.
    Chaourar, B.: A linear time algorithm for a variant of the MAX CUT problem in series parallel graphs. Adv. Oper. Res. (2017) Google Scholar
  6. 6.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1(1), 269–271 (1959)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Haglin, D.J., Venkatesan, S.M.: Approximation and intractability results for the maximum cut problem and its variants. IEEE Trans. Comput. 40(1), 110–113 (1991)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Papadimitriou, C., Yannakakis, M.: Shortest paths without a map. Theor. Comput. Sci. 84(1), 127–150 (1991)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Shiri, D., Salman, F.S.: On the randomized online strategies for the \(k\)-Canadian traveler problem. J. Comb. Opt. 38, 254–267 (2019)Google Scholar
  10. 10.
    Westphal, S.: A note on the \(k\)-Canadian Traveller Problem. Inf. Process. Lett. 106(3), 87–89 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Xu, Y., Hu, M., Su, B., Zhu, B., Zhu, Z.: The Canadian traveller problem and its competitive analysis. J. Comb. Optim. 18(2), 195–205 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.LRI, Université Paris-Sud, Université Paris-SaclayOrsayFrance
  2. 2.Bell Labs, Nokia Paris-SaclayNozayFrance
  3. 3.LTCI, Telecom ParisTech, Université Paris-SaclayParisFrance

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