  Given a set of directional visual sensors, the $k$-coverage problem
determines the orientation of minimal directional sensors so that each target
is covered at least $k$ times. As the problem is NP-complete, a number of
heuristics have been devised to tackle the issue. However, the existing
heuristics provide imbalance coverage of the targets--some targets are covered
$k$ times while others are left totally uncovered or singly covered. The
coverage imbalance is more serious in under-provisioned networks where there do
not exist enough sensors to cover all the targets $k$ times. Therefore, we
address the problem of covering each target at least $k$ times in a balanced
way using minimum number of sensors. We study the existing Integer Linear
Programming (ILP) formulation for single coverage and extend the idea for
$k$-coverage. However, the extension does not balance the coverage of the
targets. We further propose Integer Quadratic Programming (IQP) and Integer
Non-Linear Programming (INLP) formulations that are capable of addressing the
coverage balancing. As the proposed formulations are computationally expensive,
we devise a faster Centralized Greedy $k$-Coverage Algorithm (CGkCA) to
approximate the formulations. Finally, through rigorous simulation experiments
we show the efficacy of the proposed formulations and the CGkCA.
