# Lower Bounds for the Capture Time: Linear, Quadratic, and Beyond

• Klaus-Tycho Förster
• Jara Uitto
• Roger Wattenhofer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9439)

## Abstract

In the classical game of Cops and Robbers on graphs, the capture time is defined by the least number of moves needed to catch all robbers with the smallest amount of cops that suffice. While the case of one cop and one robber is well understood, it is an open question how long it takes for multiple cops to catch multiple robbers. We show that capturing $$\ell \in {\mathcal{O}}\left(n\right)$$ robbers can take $$\Omega\left(\ell \cdot n\right)$$ time, inducing a capture time of up to $$\Omega\left(n^2\right)$$. For the case of one cop, our results are asymptotically optimal. Furthermore, we consider the case of a superlinear amount of robbers, where we show a capture time of $$\Omega \left(n^2 \cdot \log\left(\ell/n\right) \right)$$.

## Keywords

Cayley Graph Graph Construction Capture Time Classical Game Search Number
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• Klaus-Tycho Förster
• 1