Approximation Algorithms for Minimum Sum of Moving-Distance and Opening-Costs Target Coverage Problem

In this paper, we study the Minimum Sum of Moving-Distance and Opening-Costs Target Coverage problem (MinMD$+$OCTC). Given a set of targets and a set of base stations on the plane, an opening cost function for every base station, the opened base stations can emit mobile sensors with a radius of $r$ from base station to cover the targets. The goal of MinMD$+$OCTC is to cover all the targets and minimize the sum of the opening cost and the moving distance of mobile sensors. We give the optimal solution in polynomial time for the MinMD$+$OCTC problem with targets on a straight line, and present a 8.928 approximation algorithm for a special case of the MinMD$+$OCTC problem with the targets on the plane.