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

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 $\Omega \left(n^2 \cdot \log\left(\ell/n\right) \right)$.