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Computing the k-resilience of a synchronized multi-robot system

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Abstract

We study an optimization problem that arises in the design of covering strategies for multi-robot systems. Consider a team of n cooperating robots traveling along predetermined closed and disjoint trajectories. Each robot needs to periodically communicate information to nearby robots. At places where two trajectories are within range of each other, a communication link is established, allowing two robots to exchange information, provided they are “synchronized”, i.e., they visit the link at the same time. In this setting a communication graph is defined and a system of robots is called synchronized if every pair of neighbors is synchronized. If one or more robots leave the system, then some trajectories are left unattended. To handle such cases in a synchronized system, when a live robot arrives to a communication link and detects the absence of the neighbor, it shifts to the neighboring trajectory to assume the unattended task. If enough robots leave, it may occur that a live robot enters a state of starvation, failing to permanently meet other robots during flight. To measure the tolerance of the system under this phenomenon we define the k-resilience as the minimum number of robots whose removal may cause k surviving robots to enter a state of starvation. We show that the problem of computing the k-resilience is NP-hard if k is part of the input, even if the communication graph is a tree. We propose algorithms to compute the k-resilience for constant values of k in general communication graphs and show more efficient algorithms for systems whose communication graph is a tree.

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Notes

  1. In practice, the trajectories need not be congruent provided they are not too different in length and a suitable range of speeds is available to the robots.

  2. Note that a path between two robots may have two arcs in the same trajectory, see Fig. 5c for an example. Therefore, distinct indexes i and j may exist such that \(C_i=C_j\). This does not affect the proof of the claim because it is not required that the arcs belong to different trajectories.

  3. Subset of nodes in a graph that does not contain two adjacent nodes.

  4. A caterpillar tree is a tree in which all the vertices are within distance 1 of a central path.

  5. \(T_{gcd}(n)=O(\log n(\log \log n)^2\log \log \log n)\) according to Stehlé and Zimmermann (2004).

  6. \(\tilde{O}\) notation hides polylogarithmic factors.

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Authors and Affiliations

  1. Department of Computer Science, University of Texas at Dallas, 800 West Campbell Road, Richardson, TX, 75080, USA

    Sergey Bereg

  2. Higher Technical School of Engineering, University of Seville, Camino de los Descubrimientos, 41092, Seville, Spain

    Luis-Evaristo Caraballo & José-Miguel Díaz-Báñez

  3. Department of Computer Science, University of Denver, 2360 South Gaylord Street, Denver, CO, 80208, USA

    Mario A. Lopez

Authors
  1. Sergey Bereg
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  2. Luis-Evaristo Caraballo
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  3. José-Miguel Díaz-Báñez
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  4. Mario A. Lopez
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Corresponding author

Correspondence to Sergey Bereg.

Additional information

A 4-page abstract of this paper appeared in the informal and non-selective workshop EuroCG17 (Bereg et al. 2017). This research has received funding from the projects GALGO (Spanish Ministry of Economy and Competitiveness, MTM2016-76272-R AEI/FEDER,UE) and CONNECT (EU-H2020/MSCA under Grant Agreement 2016-734922). Sergey Bereg was supported in part by NSF award CCF-1718994.

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Bereg, S., Caraballo, LE., Díaz-Báñez, JM. et al. Computing the k-resilience of a synchronized multi-robot system. J Comb Optim 36, 365–391 (2018). https://doi.org/10.1007/s10878-018-0297-3

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