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Fast and accurate unveiling of power-line outages is of paramount importance not only for preventing faults that may lead to blackouts, but also for routine monitoring and control tasks of the smart grid, including state estimation and optimal power flow. Existing approaches are either challenged by the combinatorial complexity issues involved and are thus limited to identifying single and double line-outages or they invoke less pragmatic assumptions such as conditionally independent phasor angle measurements available across the grid. Using only a subset of voltage phasor angle data, the present paper develops a near real-time algorithm for identifying multiple line outages at the affordable complexity of solving a sparse signal reconstruction problem via either greedy steps or coordinate descent iterations. Recognizing that the number of line outages is a small fraction of the total number of lines, the novel approach relies on reformulating the DC linear power flow model as a sparse overcomplete expansion and leveraging contemporary advances in compressive sampling and variable selection. This sparse representation can also be extended to incorporate available information on the internal system and more general line-parameter faults. Analysis and simulated tests on 118-, 300-, and 2383-bus systems confirm the effectiveness of identifying sparse power line outages.