Collaborative Delivery with Energy-Constrained Mobile Robots

  • Andreas Bärtschi
  • Jérémie Chalopin
  • Shantanu Das
  • Yann Disser
  • Barbara Geissmann
  • Daniel Graf
  • Arnaud Labourel
  • Matúš Mihalák
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9988)


We consider the problem of collectively delivering some message from a specified source to a designated target location in a graph, using multiple mobile agents. Each agent has a limited energy which constrains the distance it can move. Hence multiple agents need to collaborate to move the message, each agent handing over the message to the next agent to carry it forward. Given the positions of the agents in the graph and their respective budgets, the problem of finding a feasible movement schedule for the agents can be challenging. We consider two variants of the problem: in non-returning delivery, the agents can stop anywhere; whereas in returning delivery, each agent needs to return to its starting location, a variant which has not been studied before. We first provide a polynomial-time algorithm for returning delivery on trees, which is in contrast to the known (weak) \(\mathrm {NP}\)-hardness of the non-returning version. In addition, we give resource-augmented algorithms for returning delivery in general graphs. Finally, we give tight lower bounds on the required resource augmentation for both variants of the problem. In this sense, our results close the gap left by previous research.


Mobile Robot Planar Graph Mobile Agent Travel Salesman Problem Target Node 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was partially supported by the project ANR-ANCOR (anr-14-CE36-0002-01) and the SNF (project 200021L_156620).


  1. 1.
    Albers, S., Henzinger, M.R.: Exploring unknown environments. SIAM J. Comput. 29(4), 1164–1188 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Anaya, J., Chalopin, J., Czyzowicz, J., Labourel, A., Pelc, A., Vaxès, Y.: Convergecast and broadcast by power-aware mobile agents. Algorithmica 74(1), 117–155 (2016)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Applegate, D.L., Bixby, R.E., Chvatal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study (Princeton Series in Applied Mathematics). Princeton University Press, Princeton (2007)Google Scholar
  4. 4.
    Awerbuch, B., Betke, M., Rivest, R.L., Singh, M.: Piecemeal graph exploration by a mobile robot. Inf. Comput. 152(2), 155–172 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bärtschi, A., Chalopin, J., Das, S., Disser, Y., Geissmann, B., Graf, D., Labourel, A., Mihalák, M.: Collaborative Delivery with Energy-Constrained Mobile Robots, arXiv preprint. (2016)
  6. 6.
    Bender, M.A., Slonim, D.K.: The power of team exploration: two robots can learn unlabeled directed graphs. In: 35th Symposium on Foundations of Computer Science, FOCS 1994, pp. 75–85 (1994)Google Scholar
  7. 7.
    Betke, M., Rivest, R.L., Singh, M.: Piecemeal learning of an unknown environment. Mach. Learn. 18(2), 231–254 (1995)Google Scholar
  8. 8.
    Biló, D., Disser, Y., Gualá, L., Mihal’ák, M., Proietti, G., Widmayer, P.: Polygon-constrained motion planning problems. In: Flocchini, P., Gao, J., Kranakis, E., der Heide, F.M. (eds.) ALGOSENSORS 2013. LNCS, vol. 8243, pp. 67–82. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  9. 9.
    Chalopin, J., Das, S., Mihalák, M., Penna, P., Widmayer, P.: Data delivery by energy-constrained mobile agents. In: Flocchini, P., Gao, J., Kranakis, E., der Heide, F.M. (eds.) ALGOSENSORS 2013. LNCS, vol. 8243, pp. 111–122. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  10. 10.
    Chalopin, J., Jacob, R., Mihalák, M., Widmayer, P.: Data delivery by energy-constrained mobile agents on a line. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014, Part II. LNCS, vol. 8573, pp. 423–434. Springer, Heidelberg (2014)Google Scholar
  11. 11.
    Czyzowicz, J., Diks, K., Moussi, J., Rytter, W.: Communication problems for mobile agents exchanging energy. In: 23rd International Colloquium on Structural Information and Communication Complexity SIROCCO 2016 (2016)Google Scholar
  12. 12.
    Das, S., Dereniowski, D., Karousatou, C.: Collaborative exploration by energy-constrained mobile robots. In: 22th International Colloquium on Structural Information and Communication Complexity SIROCCO 2015, pp. 357–369 (2015)Google Scholar
  13. 13.
    Demaine, E.D., Hajiaghayi, M., Mahini, H., Sayedi-Roshkhar, A.S., Oveisgharan, S., Zadimoghaddam, M.: Minimizing movement. ACM Trans. Algorithms 5(3), 1–30 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dereniowski, D., Disser, Y., Kosowski, A., Pajkak, D., Uznański, P.: Fast collaborative graph exploration. Inf. Comput. 243, 37–49 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Duncan, C.A., Kobourov, S.G., Kumar, V.S.A.: Optimal constrained graph exploration. In: 12th ACM Symposium on Discrete Algorithms, SODA 2001, pp. 807–814 (2001)Google Scholar
  16. 16.
    Dynia, M., Korzeniowski, M., Schindelhauer, C.: Power-aware collective tree exploration. In: Grass, W., Sick, B., Waldschmidt, K. (eds.) ARCS 2006. LNCS, vol. 3894, pp. 341–351. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Dynia, M., Łopuszański, J., Schindelhauer, C.: Why robots need maps. In: Prencipe, G., Zaks, S. (eds.) SIROCCO 2007. LNCS, vol. 4474, pp. 41–50. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    Edmonds, J., Johnson, E.L.: Matching, euler tours and the chinese postman. Math. Program. 5(1), 88–124 (1973)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Flocchini, P., Prencipe, G., Santoro, N.: Distributed Computing by Oblivious Mobile Robots. Morgan & Claypool, San Rafeal (2012)MATHGoogle Scholar
  20. 20.
    Fraigniaud, P., Ga̧sieniec, L., Kowalski, D.R., Pelc, A.: Collective tree exploration. Networks 48(3), 166–177 (2006)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Ortolf, C., Schindelhauer, C.: Online multi-robot exploration of grid graphs with rectangular obstacles. In: 24th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2012, pp. 27–36 (2012)Google Scholar
  23. 23.
    Panaite, P., Pelc, A.: Exploring unknown undirected graphs. J. Algorithms 33(2), 281–295 (1999)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Andreas Bärtschi
    • 1
  • Jérémie Chalopin
    • 2
  • Shantanu Das
    • 2
  • Yann Disser
    • 3
  • Barbara Geissmann
    • 1
  • Daniel Graf
    • 1
  • Arnaud Labourel
    • 2
  • Matúš Mihalák
    • 4
  1. 1.ETH ZürichZürichSwitzerland
  2. 2.LIF, CNRS and Aix-Marseille UniversitéMarseilleFrance
  3. 3.TU BerlinBerlinGermany
  4. 4.Maastricht UniversityMaastrichtThe Netherlands

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