Distance-Constrained Line Routing Problem

On the plane, the barrier is a line segment, and the mobile sensors are initially located at some points (depots). Each sensor can travel a limited-length path, starting and ending at its depot. That part of the barrier, along which sensor moved, is covered by this sensor. It is required to find a min-power subset of sensors covering the entire barrier. The complexity of this problem is not known. In this paper, we have found the special cases of polynomial solvability and state some necessary and sufficient conditions for the existence of the solution. An efficient (polynomial) algorithm for checking the existence of the solution is proposed. Moreover, we have developed some approximation algorithms. In particular, an efficient implementation of the dynamic programming algorithm, which in some special cases yields an optimal solution, is proposed.