Boundary Patrolling by Mobile Agents with Distinct Maximal Speeds

  • Jurek Czyzowicz
  • Leszek Gąsieniec
  • Adrian Kosowski
  • Evangelos Kranakis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6942)


A set of k mobile agents are placed on the boundary of a simply connected planar object represented by a cycle of unit length. Each agent has its own predefined maximal speed, and is capable of moving around this boundary without exceeding its maximal speed. The agents are required to protect the boundary from an intruder which attempts to penetrate to the interior of the object through a point of the boundary, unknown to the agents. The intruder needs some time interval of length τ to accomplish the intrusion. Will the intruder be able to penetrate into the object, or is there an algorithm allowing the agents to move perpetually along the boundary, so that no point of the boundary remains unprotected for a time period τ? Such a problem may be solved by designing an algorithm which defines the motion of agents so as to minimize the idle time I, i.e., the longest time interval during which any fixed boundary point remains unvisited by some agent, with the obvious goal of achieving I < τ.

Depending on the type of the environment, this problem is known as either boundary patrolling or fence patrolling in the robotics literature. The most common heuristics adopted in the past include the cyclic strategy, where agents move in one direction around the cycle covering the environment, and the partition strategy, in which the environment is partitioned into sections patrolled separately by individual agents. This paper is, to our knowledge, the first study of the fundamental problem of boundary patrolling by agents with distinct maximal speeds. In this scenario, we give special attention to the performance of the cyclic strategy and the partition strategy. We propose general bounds and methods for analyzing these strategies, obtaining exact results for cases with 2, 3, and 4 agents. We show that there are cases when the cyclic strategy is optimal, cases when the partition strategy is optimal and, perhaps more surprisingly, novel, alternative methods have to be used to achieve optimality.


mobile agents boundary patrolling fence patrolling idleness 


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  1. 1.
    Agmon, N., Hazon, N., Kaminka, G.A.: The giving tree: constructing trees for efficient offline and online multi-robot coverage. Ann. Math. Artif. Intell. 52(2-4), 143–168 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agmon, N., Kraus, S., Kaminka, G.A.: Multi-robot perimeter patrol in adversarial settings. In: ICRA, pp. 2339–2345 (2008)Google Scholar
  3. 3.
    Almeida, A., Ramalho, G., Santana, H., Azevedo Tedesco, P., Menezes, T., Corruble, V., Chevaleyre, Y.: Recent advances on multi-agent patrolling. In: Bazzan, A.L.C., Labidi, S. (eds.) SBIA 2004. LNCS (LNAI), vol. 3171, pp. 474–483. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Amigoni, F., Basilico, N., Gatti, N., Saporiti, A., Troiani, S.: Moving game theoretical patrolling strategies from theory to practice: An usarsim simulation. In: ICRA, pp. 426–431 (2010)Google Scholar
  5. 5.
    Bampas, E., Gąsieniec, L., Hanusse, N., Ilcinkas, D., Klasing, R., Kosowski, A.: Euler tour lock-in problem in the rotor-router model. In: Keidar, I. (ed.) DISC 2009. LNCS, vol. 5805, pp. 423–435. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Barajas, J., Serra, O.: Regular chromatic number and the lonely runner problem. Electronic Notes in Discrete Mathematics 29, 479–483 (2007)CrossRefzbMATHGoogle Scholar
  7. 7.
    Barajas, J., Serra, O.: The lonely runner with seven runners. Electron. J. Combin 15(1) (2008)Google Scholar
  8. 8.
    Bienia, W., Goddyn, L., Gvozdjak, P., Sebő, A., Tarsi, M.: Flows, View Obstructions, and the Lonely Runner. Journal of Combinatorial Theory, Series B 72(1), 1–9 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chevaleyre, Y.: Theoretical analysis of the multi-agent patrolling problem. In: IAT, pp. 302–308 (2004)Google Scholar
  10. 10.
    Elmaliach, Y., Agmon, N., Kaminka, G.A.: Multi-robot area patrol under frequency constraints. Ann. Math. Artif. Intell. 57(3-4), 293–320 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Elmaliach, Y., Shiloni, A., Kaminka, G.A.: A realistic model of frequency-based multi-robot polyline patrolling. In: AAMAS (1), pp. 63–70 (2008)Google Scholar
  12. 12.
    Elor, Y., Bruckstein, A.M.: Autonomous multi-agent cycle based patrolling. In: ANTS Conference, pp. 119–130 (2010)Google Scholar
  13. 13.
    Gabriely, Y., Rimon, E.: Spanning-tree based coverage of continuous areas by a mobile robot. In: ICRA, pp. 1927–1933 (2001)Google Scholar
  14. 14.
    Hazon, N., Kaminka, G.A.: On redundancy, efficiency, and robustness in coverage for multiple robots. Robotics and Autonomous Systems 56(12), 1102–1114 (2008)CrossRefGoogle Scholar
  15. 15.
    Horvat, C.H., Stoffregen, M.: A solution to the lonely runner conjecture for almost all points. Technical Report arXiv:1103.1662v1 (2011)Google Scholar
  16. 16.
    Machado, A., Ramalho, G.L., Zucker, J.-D., Drogoul, A.: Multi-agent patrolling: An empirical analysis of alternative architectures. In: Sichman, J.S., Bousquet, F., Davidsson, P. (eds.) MABS 2002. LNCS (LNAI), vol. 2581, pp. 155–170. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  17. 17.
    Marey, E.J.: La méthode graphique (1878)Google Scholar
  18. 18.
    Marino, A., Parker, L.E., Antonelli, G., Caccavale, F.: Behavioral control for multi-robot perimeter patrol: A finite state automata approach. In: ICRA, pp. 831–836 (2009)Google Scholar
  19. 19.
    Pasqualetti, F., Franchi, A., Bullo, F.: On optimal cooperative patrolling. In: CDC, pp. 7153–7158 (2010)Google Scholar
  20. 20.
    Wills, J.M.: Zwei Sätze über inhomogene diophantische Approximation von Irrationalzehlen. Monatshefte für Mathematik 71(3), 263–269 (1967)CrossRefzbMATHGoogle Scholar
  21. 21.
    Yanovski, V., Wagner, I.A., Bruckstein, A.M.: A distributed ant algorithm for efficiently patrolling a network. Algorithmica 37(3), 165–186 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Jurek Czyzowicz
    • 1
  • Leszek Gąsieniec
    • 2
  • Adrian Kosowski
    • 3
  • Evangelos Kranakis
    • 4
  1. 1.Université du Québec en OutaouaisGatineauCanada
  2. 2.University of LiverpoolLiverpoolUK
  3. 3.INRIA Bordeaux Sud-Ouest, LaBRITalenceFrance
  4. 4.Carleton UniversityOttawaCanada

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