The Potts model has many applications. It is equivalent to some min-cut and max-flow models. Primal-dual algorithms have been used to solve these problems. Due to the special structure of the models, convergence proof is still a difficult problem. In this work, we developed two novel, preconditioned, and over-relaxed alternating direction methods of multipliers (ADMM) with convergence guarantee for these models. Using the proposed preconditioners or block preconditioners, we get accelerations with the over-relaxation variants of preconditioned ADMM. The preconditioned and over-relaxed Douglas-Rachford splitting methods are also considered for the Potts model. Our framework can handle both the two-labeling or multi-labeling problems with appropriate block preconditioners based on Eckstein-Bertsekas and Fortin-Glowinski splitting techniques.
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