This paper considers the problem of sparse signal recovery where there is a structure in the signal. Efficient recovery schemes can be designed to leverage the signal structure. Following the model-based compressive sensing framework, we have developed an efficient algorithm for both head and tail approximations for the model-projection problem. The problem is modeled as a constrained graph optimization problem, which is an NP-hard optimization problem. Solving the NP-hard optimization program is then transformed to solving a linear program and finding a randomized algorithm to find an integral solution. The integral solution is optimal-in-expectation. The algorithm is proved to have the same geometric convergence as previous work. The algorithm has been tested on various compressing matrices. It worked well with the matrices with the Restricted Isometry Property (RIP), also worked well with some matrices that have not been shown to have RIP. The proposed algorithm demonstrated improved recoverability and used fewer number of iterations to recover the signal.