This paper addresses the problem of approximate MAP-MRF inference in general graphical models.
Following , we consider a family of linear programming relaxations of the problem where each
relaxation is specified by a set of nested pairs of factors for which the marginalization constraint
needs to be enforced. We develop a generalization of the TRW-S algorithm  for this problem,
where we use a decomposition into junction chains, monotonic w.r.t. some ordering on the nodes.
This generalizes the monotonic chains in  in a natural way. We also show how to deal with
nested factors in an efficient way. Experiments show an improvement over min-sum diffusion,
MPLP and subgradient ascent algorithms on a number of computer vision and natural language processing problems.