  Applying the max-product (and belief-propagation) algorithms to loopy graphs
is now quite popular for best assignment problems. This is largely due to their
low computational complexity and impressive performance in practice. Still,
there is no general understanding of the conditions required for convergence
and/or the optimality of converged solutions. This paper presents an analysis
of both attenuated max-product (AMP) decoding and weighted min-sum (WMS)
decoding for LDPC codes which guarantees convergence to a fixed point when a
weight parameter, {\beta}, is sufficiently small. It also shows that, if the
fixed point satisfies some consistency conditions, then it must be both the
linear-programming (LP) and maximum-likelihood (ML) solution.
  For (dv,dc)-regular LDPC codes, the weight must satisfy {\beta}(dv-1) \leq 1
whereas the results proposed by Frey and Koetter require instead that
{\beta}(dv-1)(dc-1) < 1. A counterexample which shows a fixed point might not
be the ML solution if {\beta}(dv-1) > 1 is also given. Finally, connections are
explored with recent work by Arora et al. on the threshold of LP decoding.
