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As a promising way in surveillance applications, wireless sensor networks (WSNs) often encounter the target coverage (TC) problem, i.e., scheduling energy-limited sensors to monitor physical targets to prolong the network lifetime. Due to the complexity of the problem (scheduling in time domain), previous proposals mainly focus on heuristics and provable optimal algorithms remain unknown. In this paper, we fill in the research blank by providing several theoretical results. First, we present a mathematical formulation and several investigations of the problem in time domain. Such time-related results provide fundamental understandings of the problem and serve as a basis. Second, we offer an upper bound on the network lifetime derived from the time-dependant formulation. The bound, which is solvable in polynomial-time, serves as a performance benchmark. Third, we verify the set cover-based method, which is widely used by previous studies, via a transformation of the problem from time to space domain. Lastly, we offer a specialized nonlinear column generation (CG) based approach to solve the problem in space domain optimally. Simulation results show that not only the bound is effective, but also the CG-based approach offers significant improvement on the network lifetime over a brutal search algorithm and a state-of-art heuristic.