We consider discrete pairwise energy minimization prob-lem (weighted constraint satisfaction, max-sum labeling) and methods that identify a globally optimal partial assign-ment of variables. When finding a complete optimal assign-ment is intractable, determining optimal values for a part of variables is an interesting possibility. Existing methods are based on different sufficient conditions. We propose a new sufficient condition for partial optimality which is: (1) ver-ifiable in polynomial time (2) invariant to reparametriza-tion of the problem and permutation of labels and (3) in-cludes many existing sufficient conditions as special cases. We pose the problem of finding the maximum optimal par-tial assignment identifiable by the new sufficient condition. A polynomial method is proposed which is guaranteed to

Potts energy frequently occurs in computer vision appli-cations. We present an efficient parallel method for opti-mizing Potts energy based on the extension of hierarchi-cal fusion algorithm. Unlike previous parallel graph-cut based optimization algorithms, our approach has optimality bounds even after a single iteration over all labels, i.e. after solving only k-1 max-flow problems, where k is the number of labels. This is perhaps the minimum number of max-flow problems one has to solve to obtain a solution with opti-mality guarantees. Our approximation factor is O(log2 k). Although this is not as good as the factor of 2 approxima-tion of the well known expansion algorithm, we achieve very good results in practice. In particular, we found that the re-sults of our algorithm after one iteration are always better than the results after one iteration of the expansion algo-rithm. We demonstrate experimentally the computational advantages of our parallel implementation on the problem of stereo correspondence, achieving a factor of 1.5 to 2.6 speedup compared to the serial implementation. These re-sults were obtained with a small number of processors. The expected speedups with a larger number of processors are greater. 1.

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