${\boldsymbol~k}$-prize-collecting minimum power cover problem with submodular penalties on a plane

Abstract

Given a set of $n$ points, a set of $m$ sensors on a plane, and a positive integer $k$ ($\leq~n$).Each sensor $s$ can adjust its power $p(s)$ and covering range, which is a disk of radius $r(s)$ satisfying $p(s)=r(s)^{\alpha}$, where $\alpha\geq~1$ is called the attenuation factor of power. The $k$-prize-collecting minimum power cover problem with submodular penalties on a plane determines a power assignmentsuch that at least $k$ users are covered. The goal is to minimize the overall power of the power assignment plus the penalty of the uncovered user set, where the penalty is determined by a submodular function. This problem generalizes the well-known minimum power cover problem, minimum power partial cover problem, and $k$-prize-collecting minimum power cover problem.Based on the geometric properties of the semi-disjoint disk set and primal-dual method, we present a two-phase $(5\cdot~2^{\alpha}+1)$-approximation algorithm. When the penalty function is linear, the approximation ratio of our algorithm is at most $5\cdot~2^{\alpha}$.