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

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

Research output: Chapter in Book/Report/Conference proceedingConference contribution

## Abstract

We consider the following surveillance problem: Given a set $P$ of $n$ sites in a metric space and a set 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. We have two main results. We consider cyclic solutions in which the set of sites must be partitioned into $\ell$ groups, for some~$\ell \leq k$, 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 $1+\varepsilon$ factor loss in the approximation factor and an $O\left(\left( k/\varepsilon \right)^k\right)$ factor loss in the runtime, for any $\varepsilon >0$. 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. 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 $(2(1-1/k)+\varepsilon)$-approximation of the optimal unrestricted solution. If the conjecture mentioned above is true, then our algorithm is actually a PTAS for the general problem in the Euclidean setting.
Original language English Proceedings of the 38th International Symposium on Computational Geometry Xavier Goaoc, Michael Kerber Schloss Dagstuhl - Leibniz-Zentrum für Informatik 14 978-3-95977-227-3 https://doi.org/10.4230/LIPIcs.SoCG.2022.2 Published - 1 Jun 2022 The 38th International Symposium on Computational Geometry - Berlin, GermanyDuration: 7 Jun 2022 → 10 Jun 2022Conference number: 38https://www.inf.fu-berlin.de/inst/ag-ti/socg22/socg.html

### Publication series

Name 38th International Symposium on Computational Geometry Schloss Dagstuhl – Leibniz-Zentrum für Informatik 1868-8969

### Symposium

Symposium The 38th International Symposium on Computational Geometry SoCG 2022 Germany Berlin 7/06/22 → 10/06/22 https://www.inf.fu-berlin.de/inst/ag-ti/socg22/socg.html

## Keywords

• Approximation
• Motion Planning
• Scheduling

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