Scheduling sensor activity to prolong the network lifetime while guaranteeing coverage and connectivity is a fundamental and critical issue in handling wireless sensor networks. Although many efforts have been made in this area, none of the prior works considers the relay hop bound constraint. As a result, the existing scheduling algorithms can provide only connectivity with uncontrollable latency to the sinks and can’t be applied to delay-sensitive applications. In this paper, we are the first one coping with the scheduling problem for network lifetime maximization under the requirements of full target coverage and connectivity with bounded relay hop. We first propose an exact LP (i.e., Linear Programming) formulation to determine the optimal solution. Then, to reduce the time complexity, we develop a ${\left ({1+\epsilon }\right)}^{2}$ -approximation algorithm based on the partitioning and shifting technique. Furthermore, we propose an approximate LP formulation whose variable size is polynomial to the number of targets and sensors and whose performance ratio is $\widehat {M}$ , where $\widehat {M}$ is the maximum number of targets covered by a sensor. The experiment results show that when the number of sensors is sufficiently large, our algorithms extend the network lifetime up to 3.68 times compared to the existing approaches. Moreover, the proposed algorithms shorten time complexity significantly. Specifically, our algorithms’ time complexity is always less than 20% that of the benchmarks.