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Multi-label submodular Markov Random Fields (MRFs) have been shown to besolvable using max-flow based on an encoding of the labels proposed byIshikawa, in which each variable $X_i$ is represented by $\ell$ nodes (where$\ell$ is the number of labels) arranged in a column. However, this method ingeneral requires $2\,\ell^2$ edges for each pair of neighbouring variables.This makes it inapplicable to realistic problems with many variables andlabels, due to excessive memory requirement. In this paper, we introduce avariant of the max-flow algorithm that requires much less storage.Consequently, our algorithm makes it possible to optimally solve multi-labelsubmodular problems involving large numbers of variables and labels on astandard computer.

Memory Efficient Max Flow for Multi-label Submodular MRFs