While linear programming (LP) decoding provides more flexibility for finite-length performance analysis than iterative message-passing (IMP) decoding, it is computationally more complex to implement in its original form, due to both the large size of the relaxed LP problem and the inefficiency of using general-purpose LP solvers. This paper explores ideas for fast LP decoding of low-density parity-check (LDPC) codes. By modifying the previously reported Adaptive LP decoding scheme to allow removal of unnecessary constraints, we first prove that LP decoding can be performed by solving a number of LP problems that each contains at most one linear constraint derived from each of the parity-check constraints. By exploiting this property, we study a sparse interior-point implementation for solving this sequence of linear programs. Since the most complex part of each iteration of the interior-point algorithm is the solution of a (usually ill-conditioned) system of linear equations for finding the step direction, we propose a preconditioning algorithm to facilitate solving such systems iteratively. The proposed preconditioning algorithm is similar to the encoding procedure of LDPC codes, and we demonstrate its effectiveness via both analytical methods and computer simulation results.