An absolute phase estimation algorithm for interferometric applications is introduced. The approach is Bayesian. Besides coping with the 2pi-periodic sinusoidal nonlinearity in the observations, the proposed methodology assumes a first-order Markov random field prior and a maximum a posteriori probability (MAP) viewpoint. For computing the MAP solution, we provide a combinatorial suboptimal algorithm that involves a multiprecision sequence. In the coarser precision, it unwraps the phase by using, essentially, the previously introduced PUMA algorithm [IEEE Trans. Image Proc.16, 698 (2007)], which blindly detects discontinuities and yields a piecewise smooth unwrapped phase. In the subsequent increasing precision iterations, the proposed algorithm denoises each piecewise smooth region, thanks to the previously detected location of the discontinuities. For each precision, we map the problem into a sequence of binary optimizations, which we tackle by computing min-cuts on appropriate graphs. This unified rationale for both phase unwrapping and denoising inherits the fast performance of the graph min-cuts algorithms. In a set of experimental results, we illustrate the effectiveness of the proposed approach.