Optimal strategies for patrolling fences
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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,...,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 = α∑_i=1^k v_i for some efficiency α∈ [1/2,1). After a series of works giving better and better efficiencies, it was conjectured that the best possible efficiency approaches 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_/v_ and the number of agents k. Our bounds of α≤1/1 + 1/s and α≤ 1 - 1/2√(k) imply that in order to achieve efficiency 1 - ϵ, at least k ≥Ω(ϵ^-2) agents with a speed ratio of s ≥Ω(ϵ^-1) are necessary. 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.
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