Iterative approximate linear programming decoding of LDPC codes with linear complexity

The problem of low complexity linear programming (LP) decoding of low-density parity-check (LDPC) codes is considered. An iterative algorithm for efficient approximate solution of this problem was proposed by Vontobel and Koetter. In this paper the convergence rate and computational complexity of this algorithm are studied. In particular we are interested in obtaining a feasible vector in the LP decoding problem, with objective function value whose distance to the minimum, normalized by the block length, can be made arbitrarily small. It is shown that such a feasible vector can be obtained in linear, in the block length, computational complexity. Combined with previous results, that have shown that the LP decoder can correct some fixed fraction of errors, we conclude that this error correction can be achieved with linear computational complexity.