A Class of Solvable Consistent Labeling Problems

The structural description ansatz often used for representing and recognizing complex objects leads to the consistent labeling problem or to some optimization problems on labeled graphs. Although this problems are NP-complete in general it is well known that they are easy solvable if the underlying graph is a tree or even a partial m-tree (i.e its treewidth is m). On the other hand the underlying graphs arising in image analysis are often lattices or even fully connected. In this paper we study a special class of consistent labeling problems where the label set is ordered and the predicates preserve some structure derived from this ordering. We show that consistent labeling can be solved in polynomial time in this case even for fully connected graphs. Then we generalize this result to the “MaxMin” problem on labeled graphs and show how to solve it if the similarity functions preserve the same structure.