  A large number of problems in computer vision can be modelled as energy
minimization problems in a Markov Random Field (MRF) or Conditional Random
Field (CRF) framework. Graph-cuts based $\alpha$-expansion is a standard
move-making method to minimize the energy functions with sub-modular pairwise
terms. However, certain problems require more complex pairwise terms where the
$\alpha$-expansion method is generally not applicable.
  In this paper, we propose an iterative {\em tiered move making algorithm}
which is able to handle general pairwise terms. Each move to the next
configuration is based on the current labeling and an optimal tiered move,
where each tiered move requires one application of the dynamic programming
based tiered labeling method introduced in Felzenszwalb et. al.
\cite{tiered_cvpr_felzenszwalbV10}. The algorithm converges to a local minimum
for any general pairwise potential, and we give a theoretical analysis of the
properties of the algorithm, characterizing the situations in which we can
expect good performance. We first evaluate our method on an object-class
segmentation problem using the Pascal VOC-11 segmentation dataset where we
learn general pairwise terms. Further we evaluate the algorithm on many other
benchmark labeling problems such as stereo, image segmentation, image stitching
and image denoising. Our method consistently gets better accuracy and energy
values than alpha-expansion, loopy belief propagation (LBP), quadratic
pseudo-boolean optimization (QPBO), and is competitive with TRWS.
