Computational imaging problems are of increasing importance in domains ranging from security to biology and medicine. In these problems computational techniques based on an imaging model are coupled with data inversion to create useful images. When the underlying desired property field itself is discrete, the corresponding discrete-valued inverse problems are extremely challenging and computationally expensive to solve because of their non-convex, enumerative nature. In this work we demonstrate a fast and robust solution approach based on a new variable splitting coupled with the alternating direction method of multipliers (ADMM) technique. This approach exploits sub-problems that can be solved using existing and fast techniques, such as graph-cut methods, and results in overall solutions of excellent quality. The method can exploit both Gaussian and Poisson noise models. We exercise the method on both binary and multi-label phantoms for challenging limited data tomographic reconstruction problems.