Approximation algorithms for maximum weighted target cover problem with distance limitations

In this paper, we study approximation algorithms for the problem of maximum weighted target cover with distance limitations (MaxWTCDL). Given n targets \(T=\left\{ t_{1},t_{2},\ldots ,t_{n}\right\} \) $T=\left\{t_1,,,t_2,,,\dots ,,,t_n\right\}$ on the plane and m mobile sensors \(S=\left\{ s_{1},s_{2},\ldots ,s_{m}\right\} \) $S=\left\{s_1,,,s_2,,,\dots ,,,s_m\right\}$ randomly deployed on the plane, each target \(t_i\) $t_i$ has a weight \(w_{i}\) $w_i$ and the sensing radius of the mobile sensors is \(r_{s}\) $r_s$ , suppose there is a movement distance constraint b for each sensor and a total movement distance constraint B, where \(B>b\) $B>b$ , the goal of MaxWTCDL is to move the mobile sensors within the distance constraints b and B to maximize the weight of covered targets. We present two polynomial time approximation algorithms. One is greedy-based, achieving approximation ratio \(\frac{1}{2v}\) $\frac{1}{2v}$ in time \(O(mn^2)\) $O(m{n}^2)$ , where . The other is LP-based, achieving approximation ratio \(\frac{1}{v}(1-e^{-1})\) $\frac{1}{v}(1-e^{-1})$ in time \(T_{LP}\) $T_{\mathit{LP}}$ , where \(T_{LP}\) $T_{\mathit{LP}}$ is the time needed to solve the linear program.