In this paper, we focus on solving the linear programming (LP) problem that arises in the decoding of low-density parity-check (LDPC) codes by means of the revised simplex method. In order to take advantage of the structure of the LP problem, we reformulate the dual LP and apply the idea of Dantzig-Wolfe (D-W) decomposition method to solve the problem. Each subproblem in the D-W decomposition method is an LP over a convex polyhedral cone. We define the convex polyhedral cone as local parity-check cone and discuss its special structures. Then, we enumerate its extreme rays and use these extreme rays to design an efficient method for the general LP decoding problem. The proposed method exhibits advantages in reducing both the storage and computational requirements.