Optimal strategies for patrolling fences

A classical multi-agent fence patrolling problem asks: What is the maximum length $L$ of a line that $k$ agents with maximum speeds $v_1,\ldots,v_k$ can patrol if each point on the line needs to be visited at least once every unit of time. It is easy to see that $L = \alpha \sum_{i=1}^k v_i$ for some efficiency $\alpha \in [\frac{1}{2},1)$. After a series of works giving better and better efficiencies, it was conjectured that the best possible efficiency approaches $\frac{2}{3}$. No upper bounds on the efficiency below $1$ were known. We prove the first such upper bounds and tightly bound the optimal efficiency in terms of the minimum ratio of speeds $s = {v_{\max}}/{v_{\min}}$ and the number of agents $k$. Guided by our upper bounds, we construct a scheme whose efficiency approaches $1$, disproving the conjecture of Kawamura and Soejima. Our scheme asymptotically matches our upper bounds in terms of the maximal speed difference and the number of agents used, proving them to be asymptotically tight. A variation of the fence patrolling problem considers a circular fence instead and asks for its circumference to be maximized. We consider the unidirectional case of this variation, where all agents are only allowed to move in one direction, say clockwise. At first, a strategy yielding $L = \max_{r \in [k]} r \cdot v_r$ where $v_1 \geq v_2 \geq \dots \geq v_k$ was conjectured to be optimal by Czyzowicz et al. This was proven not to be the case by giving constructions for only specific numbers of agents with marginal improvements of $L$. We give a general construction that yields $L = \frac{1}{33 \log_e\log_2(k)} \sum_{i=1}^k v_i$ for any set of agents, which in particular for the case $1, 1/2, \dots, 1/k$ diverges as $k \rightarrow \infty$, thus resolving a conjecture by Kawamura and Soejima affirmatively.