Improved bounds for cops-and-robber pursuit

We prove that n cops can capture (that is, some cop can get less than unit distance from) a robber in a continuous square region with side length less than √ 5n and hence that n/ √ 5 +1 cops can capture a robber in a square with side length n. We extend these results to three dimensions, proving that 0.34869 · · ·n2 + O(n) cops can capture a robber in a n × n × n cube and that a robber can forever evade fewer than 0.02168 · · ·n2 + O(n) cops in that cube. Key words. Pursuit, evasion, cops and robber, lion and man, rabbit and robot AMS(MOS) subject classifications. 49N75 Under what conditions can a robber evade cops on fixed patrol routes (that is, the cops move non-adaptively, independent of the robber’s movements)? Pursuit problems have been studied for centuries, with recent results prompted by Dumitrescu, Suzuki, and Zylinski [6] who asked, among other questions, what is the maximum number of cops that a robber can evade, that is, stay at least unit distance away fro...

• Type: Journal
• Year: 2011
• Where: COMGEO
• Authors: Laurent Alonso, Edward M. Reingold
• Keywords: Applied Computing, COMGEO 2011, Robber, Single Cop, Square Region