When Patrolmen Become Corrupted: Monitoring a Graph Using Faulty Mobile Robots

  • Jurek Czyzowicz
  • Leszek Gasieniec
  • Adrian Kosowski
  • Evangelos Kranakis
  • Danny Krizanc
  • Najmeh Taleb
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9472)


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 \({\mathfrak {I}}_k^f (G)\) the maximum time that a point of the domain may remain unvisited by reliable patrolmen. The objective is to find patrolling strategies minimizing \({\mathfrak {I}}_k^f (G)\).

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 \({\mathfrak {I}}^f_k(G) = (f+1) |E| / k\), 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 of our proof is the investigation of classes of graphs minimizing idle time (with respect to the total length of edges); an example of such a class is known in the literature under the name of Kotzig graphs.


Fault tolerant Idleness Kotzig graphs Patrolling 



This work was partially supported by NSERC grants. Research on this problem was initiated at the MITACS “International Problem Solving Workshop” held on July 16–20, 2012, in Vancouver, BC, Canada. The authors would like to express their deepest appreciation for the generous support of MITACS.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Jurek Czyzowicz
    • 1
  • Leszek Gasieniec
    • 2
  • Adrian Kosowski
    • 3
  • Evangelos Kranakis
    • 4
  • Danny Krizanc
    • 5
  • Najmeh Taleb
    • 4
  1. 1.Dépt. d’informatiqueUniv. du Québec en OutaouaisGatineauCanada
  2. 2.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK
  3. 3.LIAFAInria and Université Paris DiderotParisFrance
  4. 4.School of Computer ScienceCarleton UniversityOttawaCanada
  5. 5.Department of Mathematics and Computer ScienceWesleyan UniversityMiddletownUSA

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