Approximation algorithms for distance constraint sweep coverage with base stations

In this paper, we study the minimum distance constraint sweep coverage problem (MinSDCSC), the goal of which is to find the minimum number of mobile sensors and their trajectories such that each static sensor is visited at least once by some mobile sensor every required time interval and every mobile sensor visits a base station before running out of its energy (suppose every replenishment of energy can support a continuous movement of distance D). For the case when there is only one base station, we present an asymptotically $$\frac{\alpha \beta }{\beta -2}$$ α β β - 2 -approximation algorithm for the problem on a graph with a metric distance function, and a 2-approximation algorithm for the problem on a tree metric, where $$\alpha $$ α is the approximation ratio for the traveling salesman problem and $$\beta $$ β is the ratio between D and the distance from the base station to the farthest static sensor. For the case where there are k base stations, we show that there exists an algorithm with approximation factor at most $$k\gamma $$ k γ where $$\gamma $$ γ is the approximation ratio for the problem with one base station.