Memory Efficient Max Flow for Multi-Label Submodular MRFs

Multi-label submodular Markov Random Fields (MRFs) have been shown to be solvable using max-flow based on an encoding of the labels proposed by Ishikawa, 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 in general requires
$2\;\ell ^2$
edges for each pair of neighbouring variables. This makes it inapplicable to realistic problems with many variables and labels, due to excessive memory requirement. In this paper, we introduce a variant of the max-flow algorithm that requires much less storage. Consequently, our algorithm makes it possible to optimally solve multi-label submodular problems involving large numbers of variables and labels on a standard computer.