On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem

Authors Peyman Afshani, Mark de Berg , Kevin Buchin , Jie Gao , Maarten Löffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, Hao-Tsung Yang

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Peyman Afshani
  • Department of Computer Science, Aarhus University, Denmark
Mark de Berg
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Kevin Buchin
  • Department of Computer Science, TU Dortmund, Germany
Jie Gao
  • Department of Computer Science, Rutgers University, New Brunswick, NJ, USA
Maarten Löffler
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Amir Nayyeri
  • School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, USA
Benjamin Raichel
  • Department of Computer Science, University of Texas at Dallas, Richardson, TX, USA
Rik Sarkar
  • School of Informatics, University of Edinburgh, UK
Haotian Wang
  • Department of Computer Science, Rutgers University, New Brunswick, NJ, USA
Hao-Tsung Yang
  • School of Informatics, University of Edinburgh, UK

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Peyman Afshani, Mark de Berg, Kevin Buchin, Jie Gao, Maarten Löffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, and Hao-Tsung Yang. On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We consider the following surveillance problem: Given a set P of n sites in a metric space and a set R of k robots with the same maximum speed, compute a patrol schedule of minimum latency for the robots. Here a patrol schedule specifies for each robot an infinite sequence of sites to visit (in the given order) and the latency L of a schedule is the maximum latency of any site, where the latency of a site s is the supremum of the lengths of the time intervals between consecutive visits to s. When k = 1 the problem is equivalent to the travelling salesman problem (TSP) and thus it is NP-hard. For k ≥ 2 (which is the version we are interested in) the problem becomes even more challenging; for example, it is not even clear if the decision version of the problem is decidable, in particular in the Euclidean case. We have two main results. We consider cyclic solutions in which the set of sites must be partitioned into 𝓁 groups, for some 𝓁 ≤ k, and each group is assigned a subset of the robots that move along the travelling salesman tour of the group at equal distance from each other. Our first main result is that approximating the optimal latency of the class of cyclic solutions can be reduced to approximating the optimal travelling salesman tour on some input, with only a 1+ε factor loss in the approximation factor and an O((k/ε) ^k) factor loss in the runtime, for any ε > 0. Our second main result shows that an optimal cyclic solution is a 2(1-1/k)-approximation of the overall optimal solution. Note that for k = 2 this implies that an optimal cyclic solution is optimal overall. We conjecture that this is true for k ≥ 3 as well. The results have a number of consequences. For the Euclidean version of the problem, for instance, combining our results with known results on Euclidean TSP, yields a PTAS for approximating an optimal cyclic solution, and it yields a (2(1-1/k)+ε)-approximation of the optimal unrestricted (not necessarily cyclic) solution. If the conjecture mentioned above is true, then our algorithm is actually a PTAS for the general problem in the Euclidean setting. Similar results can be obtained by combining our results with other known TSP algorithms in non-Euclidean metrics.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Approximation
  • Motion Planning
  • Scheduling


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