TY - GEN
T1 - A tight lower bound for the capture time of the cops and robbers game
AU - Brandt, Sebastian
AU - Emek, Yuval
AU - Uitto, Jara
AU - Wattenhofer, Roger
N1 - Publisher Copyright:
© Sebastian Brandt, Yuval Emek, Jara Uitto, and Roger Wattenhofer;.
PY - 2017/7/1
Y1 - 2017/7/1
N2 - For the game of Cops and Robbers, it is known that in 1-cop-win graphs, the cop can capture the robber in O(n) time, and that there exist graphs in which this capture time is tight. When κ ≥ 2, a simple counting argument shows that in k-cop-win graphs, the capture time is at most O(nκ+1), however, no non-trivial lower bounds were previously known; indeed, in their 2011 book, Bonato and Nowakowski ask whether this upper bound can be improved. In this paper, the question of Bonato and Nowakowski is answered on the negative, proving that the O(nk+1) bound is asymptotically tight for any constant κ ≥ 2. This yields a surprising gap in the capture time complexities between the 1-cop and the 2-cop cases.
AB - For the game of Cops and Robbers, it is known that in 1-cop-win graphs, the cop can capture the robber in O(n) time, and that there exist graphs in which this capture time is tight. When κ ≥ 2, a simple counting argument shows that in k-cop-win graphs, the capture time is at most O(nκ+1), however, no non-trivial lower bounds were previously known; indeed, in their 2011 book, Bonato and Nowakowski ask whether this upper bound can be improved. In this paper, the question of Bonato and Nowakowski is answered on the negative, proving that the O(nk+1) bound is asymptotically tight for any constant κ ≥ 2. This yields a surprising gap in the capture time complexities between the 1-cop and the 2-cop cases.
KW - Capture time
KW - Cops and robbers
KW - Lower bound
UR - http://www.scopus.com/inward/record.url?scp=85027286080&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ICALP.2017.82
DO - 10.4230/LIPIcs.ICALP.2017.82
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AN - SCOPUS:85027286080
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
A2 - Muscholl, Anca
A2 - Indyk, Piotr
A2 - Kuhn, Fabian
A2 - Chatzigiannakis, Ioannis
T2 - 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
Y2 - 10 July 2017 through 14 July 2017
ER -