Covering segments on a line with drones

Abstract

Covering a set of segments in a plane with vehicles of limited autonomy is a problem of practical interest. The limited battery endurance imposes periodical visits to a static base station. Typically, two optimization problems are considered: minimize the number of tours, and minimize the total traveled distance. In a general setting, the problems are NP-hard and in this letter, we study the one-dimensional version. For covering segments on a line, we design efficient solutions for both optimization problems. First, we design a greedy algorithm that is optimal for the first task, and for both tasks when only one segment is considered. Being n and m the number of segments and tours of an optimal solution, respectively, our algorithm runs in $O(m+n)$ time. For the second criterion, our solution is based on Dynamic Programming and runs in $O\left(n^{2}m\right)$ time.