Approximation Algorithms for the Min-Max Cycle Cover Problem with Neighborhoods

In this paper we study the min-max cycle cover problem with neighborhoods, which is to find a given number of $K$ cycles to collaboratively visit $n$ Points of Interest (POIs) in a 2D space such that the length of the longest cycle among the $K$ cycles is minimized. The problem arises from many applications, including employing mobile sinks to collect sensor data in wireless sensor networks (WSNs), dispatching charging vehicles to recharge sensors in rechargeable sensor networks, scheduling Unmanned Aerial Vehicles (UAVs) to monitor disaster areas, etc. For example, consider the application of employing multiple mobile sinks to collect sensor data in WSNs. If some mobile sink has a long data collection tour while the other mobile sinks have short tours, this incurs a long data collection latency of the sensors in the long tour. Existing studies assumed that one vehicle needs to move to the location of a POI to serve it. We however assume that the vehicle is able to serve the POI as long as the vehicle is within the neighborhood area of the POI. One such an example is that a mobile sink in a WSN can receive data from a sensor if it is within the transmission range of the sensor (e.g., within 50 meters). It can be seen that the ignorance of neighborhoods will incur a longer traveling length. On the other hand, most existing studies only took into account the vehicle traveling time but ignore the POI service time. Consequently, although the length of some vehicle tour is short, the total amount of time consumed by a vehicle in the tour is prohibitively long, due to many POIs in the tour. In this paper we first study the min-max cycle cover problem with neighborhoods, by incorporating both neighborhoods and POI service time into consideration. We then propose novel approximation algorithms for the problem, by exploring the combinatorial properties of the problem. We finally evaluate the proposed algorithms via experimental simulations. Experimental results show that the proposed algorithms are promising. Especially, the maximum tour times by the proposed algorithms are only about from 80% to 90% of that by existing algorithms.