  Dual decomposition provides a tractable framework for designing algorithms
for finding the most probable (MAP) configuration in graphical models. However,
for many real-world inference problems, the typical decomposition has a large
integrality gap, due to frustrated cycles. One way to tighten the relaxation is
to introduce additional constraints that explicitly enforce cycle consistency.
Earlier work showed that cluster-pursuit algorithms, which iteratively
introduce cycle and other higherorder consistency constraints, allows one to
exactly solve many hard inference problems. However, these algorithms
explicitly enumerate a candidate set of clusters, limiting them to triplets or
other short cycles. We solve the search problem for cycle constraints, giving a
nearly linear time algorithm for finding the most frustrated cycle of arbitrary
length. We show how to use this search algorithm together with the dual
decomposition framework and clusterpursuit. The new algorithm exactly solves
MAP inference problems arising from relational classification and stereo
vision.
