Simple strategies versus optimal schedules in multi-agent patrolling

Suppose that a set of mobile agents, each with a predefined maximum speed, want to patrol a fence together so as to minimize the longest time interval during which a point on the fence is left unvisited. In 2011, Czyzowicz, G\k{a}sieniec, Kosowski and Kranakis studied this problem for the settings where the fence is an interval (a line segment) and a circle, and conjectured that the following simple strategies are always optimal: for Interval Patrolling, the simple strategy partitions the fence into subintervals, one for each agent, and lets each agent move back and forth in the assigned subinterval with its maximum speed; for Circle Patrolling, the simple strategy is to choose a number r, place the r fastest agents equidistantly around the circle, and move them at the speed of the rth agent. Surprisingly, these conjectures were then proved false: schedules were found (for some settings of maximum speeds) that slightly outperform the simple strategies. In this paper, we are interested in the ratio between the performances of optimal schedules and simple strategies. For the two problems, we construct schedules that are 4/3 times (for Interval Patrolling) and 21/20 times (for Circle Patrolling) as good, respectively, as the simple strategies. We also propose a new variant, in which we want to patrol a single point under the constraint that each agent can only visit the point some predefined time after its previous visit. We obtain some similar ratio bounds and NP-hardness results related to this problem.