We formulate and analyze a fundamental downlink transmission scheduling problem in a wireless communication system, composed of a base station and a set of users, each requesting a packet to be served within a time window. Some packets are requested by several users and can be served simultaneously due to the broadcast nature of the wireless medium. The base station can choose from a set of transmission strategies, e.g., in terms of combination of data rate and coding scheme, to serve the users. Each request can be served by a subset of transmission strategies. We seek a downlink transmission scheduling algorithm generating maximum aggregated system-wide utility. In this paper, we develop approximation algorithms for the formulated downlink transmission scheduling problem in both offline and online settings. We first establish the NP-hardness of the offline scheduling problem and the approximation hardness and bound of the online scheduling problem in non-preemptive and preemptive-restart models, respectively. We then design approximation algorithms and derive the approximation and competitive ratios of our online and offline scheduling algorithms. Numerical simulations are performed to further evaluate our algorithms in a wide range of network settings.