Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In the nonreturning version we want agents to have the same “range”, hence we set their budget to \(\zeta \).
- 2.
In the nonreturning version we assign a budget of \((1+\zeta )\) to clause agents.
- 3.
We relocate a non-returning agent by adding an edge of length \((B-B_i)\).
- 4.
Non-returning clause agents can do this if they are 3-resource-augmented; and we can prevent it for \((3-\varepsilon )\)-resource-augmentation by setting \(\zeta \) such that \((3-\varepsilon )\cdot (1+\zeta ) \ll 3\) (in fact the value of \(\zeta \) will be the same as for returning BudgetedDelivery, but we will use different bounds in the proof).
- 5.
In the nonreturning version, variable agents have a budget of \(\zeta \) and clause agents a budget of \(1+\zeta \).
- 6.
For non-returning agents we use (for \(\varepsilon < 2\)) the inequalities: \(\zeta = \tfrac{\varepsilon }{6-\varepsilon }< \tfrac{\varepsilon }{4} < \tfrac{\varepsilon }{2}\) and \((6-\varepsilon ) > 2(3-\varepsilon )\). Hence a non-returning \(\gamma \)-resource-augmented clause agent has a budget of \(\gamma (1+\zeta ) = (3-\varepsilon )(1+\zeta ) = 3-\varepsilon + \tfrac{(3-\varepsilon )\varepsilon }{6-\varepsilon }< 3-\tfrac{\varepsilon }{2} < 3-\zeta = 3\cdot (1-\tfrac{\zeta }{3})\), and thus cannot transport the message via its clause node.
References
Albers, S., Henzinger, M.R.: Exploring unknown environments. SIAM J. Comput. 29(4), 1164–1188 (2000)
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)
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)
Awerbuch, B., Betke, M., Rivest, R.L., Singh, M.: Piecemeal graph exploration by a mobile robot. Inf. Comput. 152(2), 155–172 (1999)
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. https://arxiv.org/abs/1606.05538 (2016)
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)
Betke, M., Rivest, R.L., Singh, M.: Piecemeal learning of an unknown environment. Mach. Learn. 18(2), 231–254 (1995)
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)
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)
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)
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)
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)
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)
Dereniowski, D., Disser, Y., Kosowski, A., Pajkak, D., Uznański, P.: Fast collaborative graph exploration. Inf. Comput. 243, 37–49 (2015)
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)
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)
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)
Edmonds, J., Johnson, E.L.: Matching, euler tours and the chinese postman. Math. Program. 5(1), 88–124 (1973)
Flocchini, P., Prencipe, G., Santoro, N.: Distributed Computing by Oblivious Mobile Robots. Morgan & Claypool, San Rafeal (2012)
Fraigniaud, P., Ga̧sieniec, L., Kowalski, D.R., Pelc, A.: Collective tree exploration. Networks 48(3), 166–177 (2006)
Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982)
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)
Panaite, P., Pelc, A.: Exploring unknown undirected graphs. J. Algorithms 33(2), 281–295 (1999)
Acknowledgments
This work was partially supported by the project ANR-ANCOR (anr-14-CE36-0002-01) and the SNF (project 200021L_156620).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this paper
Cite this paper
Bärtschi, A. et al. (2016). Collaborative Delivery with Energy-Constrained Mobile Robots. In: Suomela, J. (eds) Structural Information and Communication Complexity. SIROCCO 2016. Lecture Notes in Computer Science(), vol 9988. Springer, Cham. https://doi.org/10.1007/978-3-319-48314-6_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-48314-6_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-48313-9
Online ISBN: 978-3-319-48314-6
eBook Packages: Computer ScienceComputer Science (R0)