Bounding an Optimal Search Path with a Game of Cop and Robber on Graphs

  • Frédéric Simard
  • Michael Morin
  • Claude-Guy Quimper
  • François Laviolette
  • Josée Desharnais
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9255)


In search theory, the goal of the Optimal Search Path (OSP) problem is to find a finite length path maximizing the probability that a searcher detects a lost wanderer on a graph. We propose to bound the probability of finding the wanderer in the remaining search time by relaxing the problem into a stochastic game of cop and robber from graph theory. We discuss the validity of this bound and demonstrate its effectiveness on a constraint programming model of the problem. Experimental results show how our novel bound compares favorably to the DMEAN bound from the literature, a state-of-the-art bound based on a relaxation of the OSP into a longest path problem.


Optimal search path Cop and robber Constraint relaxation Pursuit games 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Trummel, K., Weisinger, J.: The complexity of the optimal searcher path problem. Operations Research 34(2), 324–327 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Stewart, T.: Search for a moving target when the searcher motion is restricted. Computers and Operations Research 6(3), 129–140 (1979)CrossRefGoogle Scholar
  3. 3.
    Stone, L.: Theory of Optimal Search. Academic Press, New York (2004)Google Scholar
  4. 4.
    Netsch, R.: The USCG search and rescue optimal planning system (SAROPS) via the commercial/joint mapping tool kit (c/jmtk). In: Proceedings of the 24th Annual ESRI User Conference, vol. 9, August 2004Google Scholar
  5. 5.
    Abi-Zeid, I., Frost, J.: A decision support system for canadian search and rescue operations. European Journal of Operational Research 162(3), 636–653 (2005)CrossRefGoogle Scholar
  6. 6.
    Washburn, A.R.: Branch and bound methods for a search problem. Naval Research Logistics 45(3), 243–257 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Lau, H., Huang, S., Dissanayake, G.: Discounted MEAN bound for the optimal searcher path problem with non-uniform travel times. European Journal of Operational Research 190(2), 383–397 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Morin, M., Lamontagne, L., Abi-Zeid, I., Lang, P., Maupin, P.: The optimal searcher path problem with a visibility criterion in discrete time and space. In: Proceedings of the 12th International Conference on Information Fusion, pp. 2217–2224 (2009)Google Scholar
  9. 9.
    Sato, H., Royset, J.O.: Path optimization for the resource-constrained searcher. Naval Research Logistics, 422–440 (2010)Google Scholar
  10. 10.
    Morin, M., Papillon, A.-P., Abi-Zeid, I., Laviolette, F., Quimper, C.-G.: Constraint programming for path planning with uncertainty. In: Milano, M. (ed.) CP 2012. LNCS, vol. 7514, pp. 988–1003. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  11. 11.
    Nowakowski, R., Winkler, P.: Vertex-to-vertex pursuit in a graph. Discrete Mathematics 43(2), 235–239 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Quilliot, A.: Problème de jeux, de point fixe, de connectivité et de représentation sur des graphes, des ensembles ordonnés et des hypergraphes. Ph.D thesis, Université de Paris VI (1983)Google Scholar
  13. 13.
    Bonato, A., Nowakowski, R.: The game of cops and robbers on graphs. American Mathematical Soc. (2011)Google Scholar
  14. 14.
    Fomin, F.V., Thilikos, D.M.: An annotated bibliography on guaranteed graph searching. Theoretical Computer Science 399(3), 236–245 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Beldiceanu, N., Demassey, S.: Global constraint catalog (2014). (accessed 2015–04)
  16. 16.
    Martins, G.H.: A new branch-and-bound procedure for computing optimal search paths. Master’s thesis, Naval Postgraduate School (1993)Google Scholar
  17. 17.
    Kehagias, A., Mitsche, D., Prałat, P.: Cops and invisible robbers: The cost of drunkenness. Theoretical Computer Science 481, 100–120 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kehagias, A., Prałat, P.: Some remarks on cops and drunk robbers. Theoretical Computer Science 463, 133–147 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Komarov, N., Winkler, P.: Capturing the Drunk Robber on a Graph. arXiv preprint arXiv:1305.4559 (2013)
  20. 20.
    Bäuerle, N., Rieder, U.: Markov Decision Processes with Applications to Finance. Springer (2011)Google Scholar
  21. 21.
    Puterman, M.L.: Markov decision processes: discrete stochastic dynamic programming, vol. 414. John Wiley & Sons (2009)Google Scholar
  22. 22.
    Barto, A.G., Sutton, R.S.: Reinforcement learning: An introduction. MIT press (1998)Google Scholar
  23. 23.
    Laburthe, F., Jussien, N.: Choco solver documentation. École de Mines de Nantes (2012)Google Scholar
  24. 24.
    O’Madadhain, J., Fisher, D., Nelson, T., White, S., Boey, Y.B.: The JUNG (Java universal network/graph) framework (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Frédéric Simard
    • 1
  • Michael Morin
    • 1
  • Claude-Guy Quimper
    • 1
  • François Laviolette
    • 1
  • Josée Desharnais
    • 1
  1. 1.Department of Computer Science and Software EngineeringUniversité LavalQuébecCanada

Personalised recommendations