  We consider linear-programming (LP) decoding of low-density parity-check
(LDPC) codes. While it is clear that one can use any general-purpose LP solver
to solve the LP that appears in the decoding problem, we argue in this paper
that the LP at hand is equipped with a lot of structure that one should take
advantage of. Towards this goal, we study the dual LP and show how
coordinate-ascent methods lead to very simple update rules that are tightly
connected to the min-sum algorithm. Moreover, replacing minima in the formula
of the dual LP with soft-minima one obtains update rules that are tightly
connected to the sum-product algorithm. This shows that LP solvers with
complexity similar to the min-sum algorithm and the sum-product algorithm are
feasible. Finally, we also discuss some sub-gradient-based methods.
