  We consider the problem of maximizing the lifetime of coverage (MLCP) of
targets in a wireless sensor network with battery-limited sensors. We first
show that the MLCP cannot be approximated within a factor less than $\ln n$ by
any polynomial time algorithm, where $n$ is the number of targets. This
provides closure to the long-standing open problem of showing optimality of
previously known $\ln n$ approximation algorithms. We also derive a new $\ln n$
approximation to the MLCP by showing a $\ln n$ approximation to the maximum
disjoint set cover problem (DSCP), which has many advantages over previous MLCP
algorithms, including an easy extension to the $k$-coverage problem. We then
present an improvement (in certain cases) to the $\ln n$ algorithm in terms of
a newly defined quantity "expansiveness" of the network. For the special
one-dimensional case, where each sensor can monitor a contiguous region of
possibly different lengths, we show that the MLCP solution is equal to the DSCP
solution, and can be found in polynomial time. Finally, for the special
two-dimensional case, where each sensor can monitor a circular area with a
given radius around itself, we combine existing results to derive a
$1+\epsilon$ approximation algorithm for solving MLCP for any $\epsilon >0$.
