Approximation algorithms for the minimum power cover problem with submodular/linear penalties

Abstract

In this paper, we introduce the minimum power cover problem with submodular and linear penalties. Suppose U is a set of users and S is a set of sensors in a d-dimensional space Rd with d≥2. Each sensor can adjust its power and the relationship between the power p(s) and the radius r(s) of the service area of sensor s satisfies p(s)=c⋅r(s)α, where c>0 and α≥1. Let p be the power assignment for each sensor and R be the set of users who are not covered by any sensor supported by p. The objective is to minimize the total power of p plus the rejected penalty of R. For a submodular penalty function that is normalized and nondecreasing, we present a combinatorial primal-dual (3α+1)-approximation algorithm. For the case in which the submodular penalty function is linear, we present a polynomial time approximation scheme based on a plane subdivision technique.