### Abstract

A team of k mobile robots is deployed on a weighted graph whose edge weights represent distances. The robots perpetually move along the domain, represented by all points belonging to the graph edges, not exceeding their maximal speed. The robots need to patrol the graph by regularly visiting all points of the domain. In this paper, we consider a team of robots (patrolmen), at most f of which may be unreliable, i.e. they fail to comply with their patrolling duties. What algorithm should be followed so as to minimize the maximum time between successive visits of every edge point by a reliable patrolmen? The corresponding measure of efficiency of patrolling called idleness has been widely accepted in the robotics literature. We extend it to the case of untrusted patrolmen; we denote by Áf k(G) the maximum time that a point of the domain may remain unvisited by reliable patrolmen. The objective is to find patrolling strategies minimizing Áf/k(G). We investigate this problem for various classes of graphs. We design optimal algorithms for line segments, which turn out to be surprisingly different from strategies for related patrolling problems proposed in the literature. We then use these results to study the case of general graphs. For Eulerian graphs G, we give an optimal patrolling strategy with idleness Áf/k(G) = (f + 1)|E|/k, where |E| is the sum of the lengths of the edges of G. Further, we show the hardness of the problem of computing the idle time for three robots, at most one of which is faulty, by reduction from 3-edge-coloring of cubic graphs — a known NP-hard problem. A byproduct oclass is known in the literature under the name of Kotzig graphs.

Original language | English (US) |
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Title of host publication | Algorithms and Computation - 26th International Symposium, ISAAC 2015, Proceedings |

Editors | Khaled Elbassioni, Kazuhisa Makino |

Publisher | Springer Verlag |

Pages | 343-354 |

Number of pages | 12 |

ISBN (Print) | 9783662489703 |

DOIs | |

State | Published - 2015 |

Event | 26th International Symposium on Algorithms and Computation, ISAAC 2015 - Nagoya, Japan Duration: Dec 9 2015 → Dec 11 2015 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9472 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 26th International Symposium on Algorithms and Computation, ISAAC 2015 |
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Country | Japan |

City | Nagoya |

Period | 12/9/15 → 12/11/15 |

### Keywords

- Fault tolerant
- Idleness
- Kotzig graphs
- Patrolling

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Algorithms and Computation - 26th International Symposium, ISAAC 2015, Proceedings*(pp. 343-354). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9472). Springer Verlag. https://doi.org/10.1007/978-3-662-48971-0 _30