Replacing the l² data fidelity term of the standard Total Variation (TV) functional with an l¹ data fidelity term has been found to offer a number of theoretical and practical benefits. Efficient algorithms for minimizing this l¹-TV functional have only recently begun to be developed, the fastest of which exploit graph representations, and are restricted to the denoising problem. We describe an alternative approach that minimizes a generalized TV functional, including both l²-TV and l¹-TV as special cases, and is capable of solving more general inverse problems than denoising (e.g. deconvolution). This algorithm is competitive with the graph-based methods in the denoising case, and is the fastest algorithm of which we are aware for general inverse problems involving a non-trivial forward linear operator.