Periodic Patrols on the Line and Other Networks

We consider a patrolling game on a graph recently introduced by Alpern et al. (2011) where the Patroller wins if he is at the attacked node while the attack is taking place. This paper studies the periodic patrolling game in the case that the attack duration is two periods. We show that if the Patroller's period is even, the game can be solved on any graph by finding the fractional covering number and fractional independence number of the graph. We also give a complete solution to the periodic patrolling game on line graphs of arbitrary size, extending the work of Papadaki et al. (2016) to the periodic domain. This models the patrolling problem on a border or channel, which is related to a classical problem of operational research going back to Morse and Kimball (1951). A periodic patrol is required to start and end at the same location, for example the place where the Patroller leaves his car to begin a foot patrol.

[1]  Steve Alpern,et al.  Patrolling Games , 2011, Oper. Res..

[2]  H. Vincent Poor,et al.  Infrastructure security games , 2014, Eur. J. Oper. Res..

[3]  Nicola Basilico,et al.  Patrolling security games: Definition and algorithms for solving large instances with single patroller and single intruder , 2012, Artif. Intell..

[4]  S. Shankar Sastry,et al.  On the optimal detection of an underwater intruder in a channel using unmanned underwater vehicles , 2011 .

[5]  Andrey Garnaev,et al.  A fast infiltration game on n arcs , 1996 .

[6]  S. Alpern.  Infiltration games on arbitrary graphs , 1992 .

[7]  Moshe Kress,et al.  Models of sensor operations for border surveillance , 2008 .

[8]  Fernando Ordóñez,et al.  Security routing games with multivehicle Chinese postman problem , 2014, Networks.

[9]  Kensaku Kikuta,et al.  An Ambush Game with an Unknown Number of Infiltrators , 2004, Oper. Res..

[10]  Nicola Basilico,et al.  A Formal Framework for Mobile Robot Patrolling in Arbitrary Environments with Adversaries , 2009, ArXiv.

[11]  Kevin D. Glazebrook,et al.  A Graph Patrol Problem with Random Attack Times , 2013, Oper. Res..

[12]  Andrey Garnaev,et al.  On the infiltration game , 1997 .

[13]  F. A. Bostock,et al.  A continuous game of ambush , 1987 .

[14]  Steve Alpern,et al.  Patrolling a Border , 2016, Oper. Res..

[15]  Robbert Fokkink,et al.  The Application of Search Games to Counter Terrorism Studies , 2013 .

[16]  Philip M. Morse,et al.  Methods of Operations Research , 1952 .

[17]  Kensaku Kikuta,et al.  Technical Note - An Ambush Game with a Fat Infiltrator , 2009, Oper. Res..

[18]  P. Goutal,et al.  On the infiltration game , 1997, Int. J. Game Theory.

[19]  A. Washburn.  On patrolling a channel , 1982 .

[20]  Noemí Zoroa,et al.  Patrolling a perimeter , 2012, Eur. J. Oper. Res..

[21]  Kevin D. Glazebrook,et al.  Optimal patrol to uncover threats in time when detection is imperfect , 2014 .

[22]  William H. Ruckle,et al.  Geometric games and their applications , 1983 .

[23]  Shmuel Gal.  On the optimality of a simple strategy for searching graphs , 2001, Int. J. Game Theory.

[24]  Jurek Czyzowicz,et al.  Optimal patrolling of fragmented boundaries , 2013, SPAA.

[25]  S. Gal.  Search Games with Mobile and Immobile Hider , 1979 .